Second cohomology group for trivial group action of Z4 on V4

Description of the group
We consider here the second cohomology group for trivial group action of specific information about::cyclic group:Z4 on the specific information about::Klein four-group, i.e.,

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

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

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

Note that since cyclic over central implies abelian, all the corresponding group extensions are abelian. Equivalently, all the 2-cocycles are symmetric 2-cocycles.

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