Cessna 182T instructor pilot reported experiencing a loss of engine power in cruise. Pilot diverted and landed safely.

Narrative

The new aircraft owner was receiving training on a cross country flight between ZZZ1 and ZZZ2 airport from myself (CFI). We departed ZZZ1 at approximately XA15 on a VFR flight to ZZZ2; the weather was clear at both departure and destination. There was a low cloud layer so we climbed to 9;500 ft initially VFR on top; in communications with ZZZ Center for flight following services; the cloud layer thickened over the mountains and we climbed to 11;500ft; then we were handed off to ZZZ Approach for flight following. Nearing ZZZ airport we climbed further to 13;500 ft approximately 15 minutes before the engine problems. Around 5 minutes later we lost GPS signal and tuned up ZZZ VOR for navigation. The student was the Pilot Flying in the left seat; at approximately XB45 the engine had a significant loss of power and we were unable to maintain altitude. We accomplished the emergency checklist items with no improvement in engine performance. I was in the right seat and took control of the aircraft; established a best glide speed; and [requested priority handling] with ZZZ Approach. ZZZ Approach gave us vectors to the nearest airport; which was ZZZ; I inquired the distance to the field and was informed we were 5 miles from the field. The student continued troubleshooting the engine and began setting up navigation and communications for the landing while I flew the aircraft. We were instructed to fly a 330 heading; after establishing the heading I requested frequencies for the ILS to Runway XX and the Tower. Around 11;500ft we entered IMC; the aircraft was equipped for IMC flight and I was qualified and current for IMC conditions. We broke out of the clouds around 9000ft and had the south end of ZZZ in sight; I requested to land on Runway XY and ZZZ Approach handed us off to ZZZ Tower. I checked in with the Tower Controller that cleared us to land and notified us of calm wind conditions. We had the south end of the runway in sight through a scattered cloud layer and turned base around 1000ft AGL; we proceeded to turn final and land around the 1000ft marks without incident. The aircraft was able to taxi to the FBO under its own power; we secured the aircraft and exited the aircraft with both of us in good physical and mental condition. In the following days we examined and repaired the aircraft; I am a licensed A&P mechanic and had some assistance from the fixed based operator. We found the turbocharged induction piping from the intercooler to the carburetor had come loose. We installed a new Cessna clamp to the joint that had come loose. After several successful test run ups and a subsequent successful test flight; we reinspected the discrepancy part with no abnormalities found. We then departed ZZZ for ZZZ2 on Day 0 and landed without incident. We continued flight training for the student's competency and he departed ZZZ2 on Day 1 to arrive home without incident.