6 volt and 12 volt receivers are not the same. That's for a reason. The 6 volt socket (and Lighter) is larger than the 12 volt units. You can't physically put a 6 volt lighter into a 12 volt plug. But a 12 will fit into a 6, but sloppily.
There are only 3 variations in socket size. 2 sizes for 12 volts and 1 size for 6 volts.
As an aside, I put an inverter (6 volts to 12 volts -positive ground) in order to run 12 volt accessories like a gps and a cell phone charger for long trips.
inverter So therefore I replaced my 6 volt socket with a 12 volt socket (from a local FLAPS) and it works great.
