Flight Instructor reported another aircraft in the pattern at a non-towered airport flew in close proximity to them throughout pattern operations; including on go-arounds and each leg of the traffic pattern.

Narrative

I was in the traffic pattern at ZZZ with my student teaching landings and takeoffs on Runway X. We had landed and done a touch and go after about halfway down the runways we started our takeoff and climb out. While that was happening there was a PA-28 from a flight school who was short final and done a go around at the same time. While I was climbing roughly to about 500 to 600 ft. I had gotten notice from other instructors I work with that their ADSB showed the Cherokee directly above us during their go around within 200 feet vertically and not until right above us during their go around did they finally side step to the right to keep us in sight. They proceeded to stay in the traffic pattern [close behind us] the entire time. Within 500 feet is a near miss air collision according to the FAA. This bozo needs to learn common sense on go around procedures regardless if its at a towered or uncontrolled airport and side step during a go around when you have aircraft straight ahead of you in the upwind to maintain visual sight. Our blind spot in a C152 is above us with high wings so obviously we couldn't see the aircraft.