There's no such thing as the M16A2 Carbine, there's simply the M4A1/M4 Carbine, M16A2/A3/A4, the CAR-15, and the AR-15 Civilian version.
The AR-15 shoots semiautomatic out of the factory, and the CAR-15 is often configured to do that.
The old XM177E2 was basically the prototype for the M4 and can shoot full automatic, as can the M4.
The M16s can only shoot in semiautomatic and in 3-round bursts, with the exception of the M16A1, which was retired.