If you only want the amp on when the car is running, then you could sense when the voltage rises to the alternators running voltage. This would keep all the circuitry in the trunk ( boot ). What you would do is have a comparator sense when the voltage rises above say 13 volts, and operate a small transistor to control the remote-on line of the power amp.
If you want to do it without the car running, you will need to sense an incoming audio signal or run a new wire from an ignition line.
Have a poke around the trunk, see if there is a line for a power antenna, or electric trunk release. Either will often only have power when the ignition is on. Even if your car does not have these accessories, often the wires are there.