Flight instructor reported loss of engine power on initial climb and a return to the airport.

Narrative

During a warmup flight for a student to finish up solo time per 61.109(5) we experienced a partial power loss on the departure leg of the traffic pattern at approximately 750MSL. As the instructor onboard I took control of the aircraft and advised ZZZ Tower and advised I needed to return to land with urgency. ZZZ Tower cleared us to land on the opposite end of the runway. A left teardrop was made and an emergency landing was executed. The flight goal was to do 3 laps in the pattern at ZZZ airport for a solo warmup flight before the student would have gone out on their own. Before start; engine start; after start; run up; taxi; and takeoff checklists and actions were done with no problem. We did one lap in the pattern to a full stop taxi back as per company policy when doing solo warmup. On the second lap we requested the option to practice a go around; when executing the go around and climbing out from Runway XX at approximately 750MSL is when the rapid power loss and engine roughness occurred. I took control and executed the emergency landing. When safely on the ground I asked Tower if they needed any information from me at the moment and they advised they did not so we taxied back to the ramp and shut down. Due to heavy braking there was some tire balding but other than this the situation came to a good conclusion both pilots and plane will fly another day.