PA28 Flight Instructor reported severe engine vibration and power loss after takeoff. The Instructor immediately returned to the airport and made a safe landing.

Narrative

This is a report to explain the incident we had with a Piper Cherokee PA 28 161; Aircraft X; on this present day at ZZZ airport.We were assigned Aircraft X to perform a flight lesson. The student pilot did preflight inspection accordingly to the proper checklist/updated including fuel inspection (Quantity and taking samples from both tanks to check for contaminants); Oil inspection; and every item suggested by the checklist while I was monitoring the procedures. Everything was inspected and was in normal condition to ensure a safe flight. After that; I did my own check on fuel; oil; magnetos; landing gear; brakes; ailerons; and flaps like every flight after the student does it too with positive result for a safe flight.After the preflight and how is suggested by checklist; the student taxied to the run up area to perform the RUNUP Check; like every flight with me monitoring the procedures. After the check; everything looked normal and he taxied to the hold short line on Runway XXR and we called for take-off clearance.We got the clearance to take-off and on the take-off (UPWIND) at 400 ft. we started experiencing engine power issues where we were not able to maintain climb rate or even level flight. RPM dropped and all the cockpit was shaking due to the strong vibrations. In a few seconds; I took control of the aircraft and the student was assisting me on what he was told to do including switching fuel tank as part of the engine failure checklist. After the checklist was done; we had no response from the engine that was having hard time to maintain power output. Then I advised ATC and landed safely on Runway XYR.We stopped at the very end of the runway and we were contacted by airport personal; fire fighters; and ambulance. They asked if we needed something and we denied any assistance since we were good. After that; I sent back the student to ramp with them on one of the trucks for safety and I taxied the aircraft back to the ramp where the maintenance personal and management was waiting for us.