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

Description of the group
This article is about the second cohomology group for trivial group action where the acting group is direct product of Z4 and Z4 and the group acted upon is cyclic group:Z4. In other words, it is about the cohomology group:

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

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

The cohomology group is isomorphic to direct product of Z4 and Z4 and Z4. In other words, it is a homocyclic group of order $$64$$ and exponent $$4$$.

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 (which correspond to abelian group extensions) and the cyclicity-preserving subgroup do not generate everything, but rather, they generate a subgroup of order 32 and index two, given as two cosets of the symmetric cohomology classes, giving $$16 \times 2 = 32$$ of the 64 possible extensions. The table below shows these generalized Baer Lie rings, with each row representing a coset of the subgroup of symmetric cohomology classes and each column a coset of the subgroup of cohomology classes with a cyclicity-preserving representative. The first row and first column give the subgroup of symmetric coohmology classes and the cyclicity-preserving subgroup respectively. Note that to avoid too many columns, repeated columns are simply indicated by specifying how many columns there are of the type:

Here now is the full table with all cohomology classes, including the ones that are not cyclicity-preserving: