G450 First Officer reported a Lateral Control Hyd Off indication during climb on a test flight for said indication. Crew followed checklist and indication reappeared. Crew returned to departure airport.

Narrative

We were performing a maintenance check flight from ZZZ to confirm correct flight-control operation; specifically the lateral axis; due to a previous discrepancy involving an amber Lateral Control Hyd Off" Crew Alerting System (CAS) message. We completed a briefing with company personnel outlining the flight requirements and conducted normal startup procedures. We then taxied to Taxiway 1 and burned approximately 4;000 pounds of fuel prior to takeoff per maintenance guidance.The takeoff roll was normal and as briefed. At 400 feet AGL; Flight Level Change was selected and the autopilot was engaged. During the ZZZZZ departure; at approximately 520 feet; the aircraft initiated the turn as published on the SID; at which point the amber 'Lateral Control Hyd Off' message illuminated. Flight controls became heavy; and the autopilot struggled to maintain lateral control; operating in manual reversion. The captain (PF) stated that he wanted me (PM) to advise Departure that we would be returning immediately. I responded that although controllability was reduced; the aircraft was still operating in a manner that did not require an immediate return; and that checklist procedures should be consulted once at a safe altitude.We continued climbing per the SID and ATC instructions. I retrieved the electronic checklist and navigated to the amber messages list for 'Lateral Control Hyd Off'. This directed us to section 3B-12-10; where we verified that the Lateral Control Switch was in the ON position; which it was. I then followed the hyperlink to checklist 03-13-30; 'Lateral Hard Over Prevention System (HOPS) Activation'.At this point; ZZZ [TRACON] was vectoring us and issuing climb and heading instructions. I reminded the captain of the degraded controllability and cautioned that the autopilot would likely struggle in any regime other than straight-and-level flight. I confirmed each checklist name verbally with the captain to ensure we agreed on the appropriate course of action. Next; I read aloud the introductory note and proceeded to Step 1: Autopilot -- Disconnect. Although the captain did disengage the autopilot at this point; he attempted to re-engage it multiple times throughout the remainder of the flight. I felt that this was improper and that hand-flying was the safest course of action; however; my objections to re-engaging the autopilot were not heeded.Continuing through the checklist; we reached the decision point: "Is system deemed necessary for continued safe flight and landing?" We both agreed that it was; and proceeded with resetting the circuit breaker. This action restored hydraulic pressure; extinguished the CAS message; and we operated normally for approximately 2-3 minutes. Given that this was a maintenance check flight; we discussed and agreed to continue with the original plan to climb to 15;000 feet and maneuver the aircraft through various configurations to verify proper flight-control operation.The next 20 minutes of flight were uneventful. At 15;000 feet; we conducted turns to headings and fixes with the autopilot both engaged and disengaged. We then slowed and configured the aircraft for landing and repeated the same maneuvering sequence. No control issues were noted; the CAS message did not reappear; and we concluded that the earlier indication had been isolated and corrected. We began our return to ZZZ; briefed the approach; completed performance initialization for landing; and accomplished the appropriate checklists. We had a discussion about how to handle the 'Lateral Control Hyd Off' message if it were to appear while on approach into ZZZ.While being sequenced onto the final approach course for the RNAV Runway XX at ZZZ; during another right turn--similar to the initial turn after departure--and in the turbulence common at low altitudes in the City X area; the "Lateral Control Hyd Off" message illuminated again. The PF struggled to maintain lateral control to the desired fix and was unable to meet the altitude constraints of the approach. With the aircraft partially configured for landing; we determined that we would be unable to stabilize the approach from our current position. We advised ATC that we needed to turn west and be re-sequenced. ATC issued an assigned altitude and heading; which was read back correctly. The captain initiated the turn; and we began accelerating and de-configuring the aircraft.While I was head-down changing flap and gear configurations; ATC reiterated the assigned altitude and heading. I looked up at the right-seat Primary Flight Display and observed that we were approximately 400 feet off altitude; trending further away from the clearance; and more than 30 degrees off the assigned heading. Due to the inability to maintain assigned altitude and heading; combined with crossing traffic 1;000 feet above us descending into ZZZ1; I immediately requested priority handling to remove altitude constraints and help absolve of us of our inability to meet the clearance. I was reprimanded in the moment for not consulting the PF prior to requesting; but I stated that the inability to comply with ATC clearances following an unstable approach and with degraded primary flight control capability warranted the request.Over the next several minutes; we coordinated with ATC regarding our intentions; aircraft condition; fuel state and passenger count. We requested a long final to allow for stabilization and executed the plan. During the final approach segment; the aircraft again became unstable; and we discontinued the approach. With a crosswind exceeding 50 degrees off the runway heading; the approach was extremely challenging. I advocated diverting to either ZZZ1 or ZZZ2 due to increased runway length and potentially more favorable wind conditions. The captain questioned making an attempt at landing the shorter runway at ZZZ; but ultimately agreed when I strongly recommended against landing on a shorter runway with a crosswind component exceeding 30 degrees.We discontinued the approach; advised ATC of our diversion to ZZZ2; and proceeded accordingly. After reprogramming the FMS; I revisited checklist 03-13-30 (Lateral HOPS Activation). Although we had previously completed it; the flight conditions had changed; and I wanted to ensure no steps had been overlooked. The captain agreed. Upon restarting the checklist; he asked whether we should reset the C4 POP circuit breaker again. I advised that the checklist specified "Pull/Reset ONCE;" and that this action had already been completed. We agreed not to reset it again and continued with the remaining steps; confirming that the system had not reset.Upon completion; the checklist directed us to brief a Flaps 20 landing; activate the GPWS override; and expect no ground spoilers upon rollout. I referenced the Inoperative Equipment Table in the Performance Handbook to determine landing distance with Flaps 20 and inoperative ground spoilers. We briefed the landing at ZZZ2 and conducted a 20-mile final with a shallow descent rate to minimize over-controlling and over-correction.After landing; we taxied clear of the runway and contacted our operations department for further guidance.I was dissatisfied with several aspects of the event. First; I felt the PF over-relied on the autopilot. Without hydraulic assistance; the system could not effectively turn or roll out on heading; resulting in frequent overshoots. This repeatedly triggered AUTOPILOT 1 & 2 FAIL and Flight Guidance Computer 1 FAIL messages. Once these messages extinguished; the autopilot was re-engaged; only to fail again on the next turn. This cycle generated multiple CAS messages and aural alerts; degrading situational awareness and requiring excessive manual correction.Additionally; spoiler fluttering was a concern. With no hydraulic pressure to hold the spoilers down; low pressure over the wing caused them to rise; triggering spoiler indications on the synoptic pages and CAS messages. I had to repeatedly remind the captain that the spoilers were not actually deployed but floating due to lack of hydraulic pressure. The resulting red Aircraft Configuration and Ground Spoilers Deployed CAS messages; along with non-mutable triple chimes; created distraction and tunnel vision. I was disappointed by what appeared to be a lack of foundational understanding of the aircraft's fail-down state. On multiple occasions; the PF attempted to deploy flight spoilers to slow the aircraft. I reminded him that the spoilers were inoperative and that drag could only be increased through flap or gear extension; or by reducing thrust.Overall; the event resulted in a positive outcome. However; it highlighted several areas that warrant further debrief with the other crew member. I would like to ensure that communication or language barriers were not contributing factors during moments when it felt as though the crew was not aligned.Cause: Mechanical Issue previously seen on this aircraft within the last 12 months.Further factors contributing to confusion inside of the mechanical issue were communications and language barriers between crew members; potentially inadequate aircraft systems knowledge; improper reliance on auto flight systems.Suggestions: Proper Maintenance; and procedures could potentially prevent this issue; but there is no evidence to suggest steps were not followed. During the event; as a crew; better communication skills; better understanding of aircraft systems; proper monitoring of flight path and energy state; better manual flying proficiency and instrument scanning / cross checking."