Light aircraft pilot reported an NMAC in the pattern at 0L7 airport.

Narrative

While completing the second loop of a right pattern on Runway 20R at Jean Airport 0L7; we were communicating on the CTAF with the one other plane in the pattern; Aircraft Z. Both our plane and the other plane were calling out each leg with our position on the published CTAF 122.9 as is typical for training at that airport. Immediately after the second landing in the pattern we heard a plane pass almost directly overhead. Looking up; we could see Aircraft Y pass directly overhead less than 100 feet over us. This was not the same aircraft that was in the pattern; as that plane was in either a right crosswind or right downwind for Runway 20R according to its last call. Aircraft Y didn't announce that it was flying straight into the pattern; or that it was doing a go around. Aircraft Y flew directly south rather than turning crosswind; then entered a long downwind right pattern for 20R. Aircraft Y didn't make any radio calls until it was on final for pattern. After landing; Aircraft Y exited the runway mid-field; then turned around and announced that it was taking off midfield despite the fact that we were holding short at the entrance to 20R waiting on landing traffic. Aircraft Y took off mid-field; without any apparent short field technique; and from our position it appeared to barely clear the end of the runway by less than 50'. We left the pattern immediately after as we determined it wasn't safe in the pattern with such a reckless pilot of Aircraft Y and returned to HND. Upon inquiry at HND the pilot of Aircraft Y that day was Person A; who according to the FAA pilot registry only appears to have a student license; and there was no instructor visible or heard in Aircraft Y that day.