C-172 Flight Instructor reported a NMAC event during initial climb with a flight of two Bonanza 35 departures. The flight of two turned early into a collision course which necessitated the C-172 to execute an evasive maneuver to avoid a collision.

Narrative

At XA:30z time; my student and I performed a touch and go at ZZZ on the North runway; on initial climb out we decided the airspace was too congested for training and asked ZZZ Tower for a departure back to ZZZ1. Tower approved us to make a straight out departure to ZZZ1 and Frequency change was approved. My student bugged runway heading and we started our climb to 3;500 ft. MSL. While getting ZZZ1 ATIS; we observed a pair of V tail airplanes in formation on their climb out of the South runway approximately 100 ft. below our altitude(3;000-3;200 ft. at the time). I kept my eye on the traffic during our climb and while passing over [a road] I observed the flight of two making a left turn to the north that intercepted our flight path. I took controls from my student and immediately started a full power climb to 3;500 ft. MSL to avoid conflict. The flight passed underneath us at less than 500 ft. After we were clear of the aircraft and level at 3;500 ft. MSL and exiting the airspace still on our assigned straight out departure I contacted ZZZ Tower again to ask where; if any miscommunication had taken place. ZZZ Tower informed me that both myself and the formation flight were cleared for straight out departures and they had no authority over where they turn once they are outside their airspace. I said thank you and left the frequency again as to not cause any conflict; however where the flight of two had turned was about 1 mile inside of the ZZZ airspace where ZZZ Tower had approved both to make a straight out departure. I believe the factor that caused this situation was the flight of two aircraft that turned on course before exiting the airspace when being cleared straight out.