Second cohomology group for trivial group action of S5 on Z2

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

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

where $$G$$ is symmetric group:S5 (the symmetric group of degree five) and $$A$$ is cyclic group:Z2. Note that $$G$$ has order 120 and $$A$$ has order 2, so all the corresponding extensions have order $$120 \times 2 = 240$$.

The cohomology group itself is isomorphic to the Klein four-group.

Computation of the group
The group can be computed using group cohomology of symmetric group:S5. As per this, we have:

$$H^2(G;A) \cong A/2A \oplus \operatorname{Ann}_A(2)$$

For $$A$$ cyclic of order two, both $$A/2A$$ and $$\operatorname{Ann}_A(2)$$ are cyclic of order two.

Elements
Note that in all cases, the base of the group extension (isomorphic to cyclic group:Z2) equals the center of the group extension and hence is a characteristic subgroup.