arm needed or solenoid to move forward three cm and keep a button pressed down until our command tells it to release
The typical control method is to control whatever it is that the button activates.
If the button is mechanical, then the command signal drives a pneumatic or electric actuator or even a mechanical solenoid which in turn drives linkage to accomplish the same task as the button movement.
If the button is electrical, then the command signal drives the coil of a relay. The relay’s contacts take the place of the switch in the circuit.
If you really need a mechanical solenoid, you can Google vendors for solenoids, which are typically available from electronic component distributors.