Second cohomology group for trivial group action of V4 on direct product of Z4 and Z2

Description of the group
We consider here the second cohomology group for trivial group action of Klein four-group on direct product of Z4 and Z2, i.e.,

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

where $$G \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_4 \times \mathbb{Z}_2$$.

The group is isomorphic to elementary abelian group:E64.

Elements
We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

Generalized Baer Lie rings
For this particular choice of $$G$$ and $$A$$, the symmetric cohomology classes (corresponding to abelian group extensions) and the cyclicity-preserving subgroup generate the whole group, i.e., we have:

$$H^2(G,A) = H^2_{sym}(G,A) + H^2_{CP}(G,A)$$

as an internal direct sum. In particular, every extension has a generalized Baer cyclicity-preserving Lie ring. A pictorial description of this would be as follows. Here, each column is a coset of $$H^2_{CP}(G,A)$$ and each row is a coset of $$H^2_{sym}(G,A)$$. The top left entry is the identity element, hence the top row corresponds to abelian group extensions and the left column corresponds to cyclicity-preserving 2-cocycles.

To avoid unnecessary duplication, we have compressed the table as follows: for each repeated column we have simply indicated the number of repeats in parentheses. The full table has 16 columns.