General Aviation Flight Instructor with student reported an NMAC at 7A3 non-towered airport. The other aircraft executed a go-around.

Narrative

While conducting pattern operations at 7A3 we where left downwind for runway 24 when Aircraft Y made a 4 mile final call. There was RNAV traffic on a 1-2 mile final and Aircraft Y caught up to them; cut across final and mid field and entered a right downwind then executed a 180 to be facing the correct direction for downwind. They where just finishing their 180 by the time we turned base and where just past mid field. They turned final behind us. We radioed that we would be touch and go. Aircraft Y radioed full stop. After climbing 200' on the touch and go I noticed Aircraft Y was right over us at 200' above us. They had gotten too close to us on final and executed a go-around. They made no radio call; did not side step; and where climbing at an extremely shallow rate. They over took us from above banked hard into the left downwind directly in front of us and flew direct to the final again ignoring the pattern. Their was another aircraft short final that Aircraft Y was 1/4 mile behind on a 1 mile final. They had to execute S turns and eventually a 360 on short final to maintain spacing. During all of this Aircraft Y was ignoring the pattern and cutting across the field and speed was well over 100 kts on final following aircraft going 60 at close distances.