Student Pilot with Flight Instructor reported after takeoff an aircraft took off immediately after them and made an unauthorized early turn resulting in a NMAC.

Narrative

I the Student Pilot had just completed a touch and go. I was on my climb out to pattern altitude and had just completed my left turn for the downwind leg to enter a left base to Runway XX for the option. Tower was in the process of training a new Controller and I think there was some confusion with call signs with the plane behind us (solo instructor) and our plane (student and instructor). Not having spoken to the other pilot (solo instructor) I think he was unaware of us being in-front of him. He took off after us and after reviewing Flightradar24 he initiated his left crosswind turn immediately after crossing the runway threshold and at an altitude of XXX MSL. But not within XXX ft. of pattern altitude that was accepted at our airport. This caused him to put himself on a collision course with us since he essentially cut us off in the pattern. My Instructor abruptly yelled 'traffic' grabbed the controls applied full power and a steep climb to the right. Right after he said traffic I looked down and to the left and saw the traffic less than 1/10th of a mile and less than 200 ft. below us.