Student pilot reported a near miss while in the traffic pattern at a non-towered airport. The Instructor took control and maneuvered to avoid a collision.

Narrative

Incident occurred at XA:50 on Day 0; my instructor and I were doing pattern work at BXK when we had a pattern conflict with a student solo. My instructor was quick to take control and performed evasive maneuvers to avoid a direct inflight collision with the other aircraft. We were in the pattern for approximately 30-45 minutes conducting training with touch and goes. The other aircraft involved was a student solo doing full stop taxi backs. We were aware of the student solo (Aircraft Y) in the area and were quick to accommodate and were fully alert on their position to ensure safety of flight. We were aware that Aircraft Y was holding short of Runway 35 as we were coming in to land. The event occurred on our 5th pattern. We were positioned on the left crosswind turning left downwind Runway 35. An alert was triggered on our PFD (Primary Flight Display) with a traffic warning indicating Aircraft Y was heading directly towards us about 90 degrees to our left wing. Aircraft Y had taken off right after us and had turned left crosswind early and headed directly towards our aircraft. My instructor quickly took control and did a steep turn to the right and avoided an inflight collision with Aircraft Y. My instructor then made a call to traffic that we performed the evasive maneuver to avoid pattern conflict and will continue to follow behind Aircraft Y instead.I believe a mixture of situational awareness; task saturation; and communication has caused such a close encounter. The event took place during a critical phase of flight where lots of tasks were being completed in a short amount of time. The combination of Aircraft Y turning left crosswind too early; cutting off Aircraft X; and the task saturation during take-off reducing situational awareness should have been the primary reasoning behind this close call. I am deeply grateful to my instructor for taking control and expertly maneuvering us away from the colliding aircraft; ensuring our safety through his exceptional CRM training. His quick actions made all the difference; and without them; we could have faced a much more dangerous situation.