General aviation Flight Instructor reported a NMAC with a jet while in the pattern.

Narrative

Aircraft Y came from 20 miles south of the airport and reported they were 20 miles downwind at 21 out of 7000ft. The next call was Aircraft Y asking how we were going to do this (How he was going to get into the pattern). Aircraft Y entered the pattern; cutting off another aircraft that was on crosswind. We had extended our downwind for another aircraft on final; and Aircraft Y was now behind us. After passing the aircraft on final; we started turning base. At the same time; Aircraft Y starts to also turn base less than a mile behind us 300ft above us also heading straight for the aircraft on final. We asked Aircraft Y if he was aware of our position on final since he was so close to us. They said yes; and that's what they had been trying to figure out earlier. We had to continue on base and overshoot the final approach path to create spacing between us and Aircraft Y. They ended up having to breakoff and go around because they did not have space between the other aircraft in front of them. They then said they were going to fly a downwind for 3 but instead just circled above the airport at 500agl all the way back around to final we had already turned on to final and he was now coming behind us. During the circle maneuver they cross directly below another aircraft in the downwind with less than 300ft vertical separation; and that other aircraft then had to take evasive action climbing and turning away from Aircraft Y.