General Aviation instructor pilot reported a near miss with another aircraft while maneuvering into the traffic pattern at a non-towered airport. The instructor took the controls and maneuvered to avoid the aircraft.

Narrative

Went with a student up to Heber to practice short field landings for stage check. We made standard calls over Deer Creek stating our position and intentions all the way in. While we were about to cross over the north end of the Deer Creek Reservoir; there was an aircraft that showed up on ADS-B departing off of Runway 22 at HCR with no radio calls. The aircraft turned toward us at the same altitude; we made two more calls stating our position and intention with no response. I took the controls from the student and entered a steep descent and he turned to follow us briefly before turning away. Still no calls made from the other aircraft. After performing 4 touch and go landings at the airport in left traffic; we decided we would do one more and while we were on base I could see the same aircraft doing aerobatics just west of the airport over the town of Charleston/midway. Once we joined final I saw them enter a high right downwind (approx. 1700' AGL); again with no calls. By the time we had done our landing and started our takeoff roll; I checked our MFD (Multi-function Flight Display) again and saw the aircraft on short final directly behind us at 200' AGL. The operator then made their only call of the day 'Aircraft Y overflying 22' and I saw the aircraft at an approximate 45 degree bank with extreme speed 100 feet above us directly parallel the runway before making a sharp turn right to re enter the downwind. We then departed southwest and had no more calls from the aircraft after multiple attempts to call them up by their tail number specifically.