Instructor pilot reported that during a touch and go training evolution; when flaps up was verbalized; the student raised the landing gear selector lever and the aircraft settled on the runway.

Narrative

Upon landing on a touch and go on runway XX at ZZZ airport; I verbalized 'flaps up'; but the student immediately raised the gear lever. Upon seeing this I immediately reached for the landing gear lever and returned it to the down position; however they had brought the lever half way up before I could stop them. The nose gear then collapsed onto the runway and the aircraft came to a stop on the center line. I then secured the aircraft and all occupants vacated the aircraft safely with no injures.I believe what causes the issue was the student's lack of experience and apparent confusion about who was controlling the flaps and landing gear. Factors that I feel could prevent a recurrence:Don't practice touch and goes in a complex multi engine aircraft. Be more clear as to who will be operating what controls because the student was instructed to feel the movements of the yoke only during the touch and go and essentially 'watch and learn'. On second thought; I would also be more clear on what the student should not operate. Additionally; more ground training on the operations of a complex aircraft and landing gear lever / system could be given to ensure the risks involved in such operations and how to mitigate them during flight training.