Cessna 150 Instructor Pilot departing a non-towered airport reported an NMAC with another aircraft performing acrobatic maneuvers over the airport.

Narrative

Doing pattern work at ZZZ with a student; we had performed a full stop landing then taxied back to runway XX via 1 to then take off again. We then visually cleared the downwind; base; and final legs with the only target in the area 3;000 feet above the airport on ADS-B in. There were no radio calls from other aircraft in the area that we heard/can remember. Having taxied onto the runway after making a radio call and started our takeoff roll; i glanced again at my Ipad; checking the target far above us which had descended rapidly to 1000 feet above. As we lifted off the target closed very close to us and behind; within about three hundred feet vertically and just behind us. I turned to see if I could see the other aircraft and was unable. I took controls from my student and kept us low to avoid potentially flying upwards into the other aircraft's flight path. Glancing at the Ipad again; I saw they pulled a sharp 360 over the approach end of the runway; low; presumably once they saw us. Again no radio calls from the other aircraft. Upon establishing in downwind; I saw what I believed to be an aerobatic aircraft taxing away from the runway.I believe that the other aircraft did some sort of aerobatic rapid descent to land on runway XX; unannounced in the ZZZ traffic pattern.