its not the fuse or the ground cable... this is really common in big bass.. my car did the same thing.. when the headunit starts cutting off that meens that your alternator cant supply enough electricity... what u want to get is a Capacitor.. this stores electricity and uses it in big bass... they are pretty expensive.. u see i have 2 12" jl audio subs i put 2 small Capacitor because if it si one big it takes way too long to charge 2 small charge alot faster..
they look just like this one on the link...
Capacitor


anyways put those on and you wont have this problem again!