Small aircraft Flight Instructor reported a rough running engine during initial climb and returned to safe landing at departure airport.

Narrative

During departure shortly after lift off the aircraft seemed to be climbing slowly. At about 300 ft. there was a noticeable shutter and the aircraft nose dropped a bit. I told the student to apply full power. We regained engine power and continued the climb with the intention of discontinuing the flight and returning to land from the pattern. As we reached 700 ft. the engine shuttered again this time almost shutting down completely. I stated 'my flight controls' to the student and took over. I verified mixture and throttle where both full forward and pushed the nose down to prevent a stall. I then immediately reported engine roughness to the Tower and began returning to land immediately. Tower asked which runway I would like. I requested to land on the nearest Runway XX and was told to make left traffic. We landed on Runway XX and came to a full stop. The engine continued to run extremely rough on the ground but did not seize. We where able to taxi back to the ramp without further incident. [There were] no damage or injuries. No indication of an engine problem was identified during the run up or taxi before take off. Proper checklist usage was adhered to and oil pressure; oil temp; fuel flow and vacuum gauges all showed in the green prior to take off roll.