There should be little or no current flow in the ground wire, so a large size, while OK, is not necessary.
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Where`d you get this 1/2 brained idea?
You would be surprised how many cable companies out there are selling home made power cords, cables, that are not UL listed. Making your own is one thing, but selling them is another.
Wire that's rated for 15-amps must have a No. 14 ground and wire rated for 20-amps must have a No. 12 ground. Eldartford, even though you are correct that no current flows through the ground, it's purpose is to act as a secondary path back to the panel in the event of a fault. This is so that the circuit breaker can trip, protecting the faulted circuit. If you cannot reset a breaker, then you have a fault.
I don't know how you determined that the size is 20AWG. Even low voltage control wires are a minimum of 18AWG. Now, if the cable has a metal jacket, then that jacket is the ground and is perfectly ok. The thin wire, in this particular case, would be a bonding wire which completes the connection to the jacket ground. If the jacket is non-metallic, then anything less than 14 wire is not safe and in most jurisdictions illegal.
Check with an electrician or the local building department to be sure.