Transistor Switching
If you did manage to switch the transistor on, It would only output 4.3V, the rest would be dissipated in the transistor. If you managed to switch 4A this would dissipate 96W - and the transistor would melt.

A MOSFET would be a better option and could be easily switched by the Arduino. (I regularly switch 10A using a MOSFET and Arduino.)

This assumes you can put the MOSFET in the negative lead. If not you should use a P-Channel MOSFET, but you need a more complex circuit to drive it.

Have you considered using a relay module?