Nontrivial Central Extension

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First we consider the fact that A_4 has a normal series Z_2 x Z_2/Z_3 and actually we can rewrite it as 1-->Z_2 x Z_2-->A_4-->Z_3-->1. We define the operator f: A_4-->Z_3 to be a homomorphism and then consider:
G={(a,b)in A_4xZ_6; f(a)=b'}
Where b'in Z_3 is the equivalent of b in Z_6 mapped to Z_3 (converting from mod 6 to mod 3). Now, we can see that G can be defined to have the same properties provided we consider the map G-->A_4 as h(a,b)=a and then the map Z_2-->G as k(0)=(e,0) and ...