Second cohomology group for trivial group action of A4 on Z2

Description of the group
This article describes the second cohomology group for trivial group action:

$$\! H^2(G;A)$$

where $$G$$ is alternating group:A4 (i.e., the alternating group on a set of size four) and $$A$$ is cyclic group:Z2.

The cohomology group is isomorphic to cyclic group:Z2.

Computation in terms of group cohomology
By the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization, we have that:

$$H^2(G;A) \cong \operatorname{Ext}^1(G^{\operatorname{ab}},A) \oplus \operatorname{Hom}(H_2(G;\mathbb{Z}),A)$$

For $$G$$ the alternating group of degree four, we have, by group cohomology of alternating group:A4, that $$G^{\operatorname{ab}}$$ (the abelianization, also the first homology group) is cyclic group:Z3 and $$H_2(G;\mathbb{Z})$$ (the Schur multiplier) is cyclic group:Z2. Plugging in, we get:

$$H^2(G;A) \cong \operatorname{Ext}^1(\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/2\mathbb{Z}) \oplus \operatorname{Hom}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}$$

Size information
We first give some quantitative size information if we use non-normalized cocycles and coboundaries: