C172 Flight Instructor reported landing safely after experiencing a substantial power loss shortly after takeoff.

Narrative

An engine failure on takeoff resulting in overrunning the runway occurred on a dual training IFR flight to ZZZ. We shot the ILS XX into ZZZ for a touch and go and everything ran good up to this point and it was a great normal landing. We retracted flaps; turned off carb heat; and then applied full power. The student then rotated at 65 and began climbing out at 85 mph. We reached an altitude of apx 200 feet and the engine immediately started running rough; it had an uneven shake and knocking sound with a large reduction in power. At this moment as the instructor I took over flight controls to pitch down to see how much runway was remaining and felt that we were not climbing anymore so I pulled the power to idle. As soon as I got the runway in sight I noticed that we did not have enough to land on it without overrunning. The engine was still running at this time so I made sure the mixture was rich and fuel on both and tried power again briefly to see if the engine came back into normal. It did not and made loud banging noises with little power so I pulled the throttle to idle again and started a full forward slip to land to try and touch down on the runway before it ended. I advised Tower and said we had a failure and would be landing and overrunning the runway. We landed with roughly 200 feet of pavement remaining with full braking and over ran into the grass beyond the threshold. I steered the aircraft to the right to avoid the approach lights and we came to a stop safely in the grass with no damage to the airplane; airfield or property of any kind. It is suspected at this time a cylinder failure cause the engine failure.