Second cohomology group for trivial group action of V4 on Z8

Description of the group
We consider here the second cohomology group for trivial group action of the specific information about::Klein four-group on specific information about::cyclic group:Z8, i.e.,

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

where $$G \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_8$$.

The cohomology group is isomorphic to elementary abelian group:E8.

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

Under the action of the automorphism group of the Klein four-group
By pre-composition, the automorphism group of the Klein four-group acts on the second cohomology group. Under this action, there are four orbits, corresponding to the four group extensions given above. Specifically, the 3 cohomology classes that give direct product of Z16 and Z2 are in one orbit, while the 3 cohomology classes that give M32 are in another orbit.

Under the action of the automorphism group of the cyclic four-group
The automorphism group of the base group has no effect on the cohomology classes. This is because this automorphism, the inverse map, pulls back trivially to the acting group, which has exponent two.

General background
We know from the general theory that there is a natural short exact sequence:

$$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0$$

where the image of $$\operatorname{Ext}^1$$ is $$H^2_{sym}(G;A)$$, i.e., the group of cohomology classes represented by symmetric 2-cocycles. We also know, again from the general theory, that the short exact sequence above splits, i.e., $$H^2_{sym}(G;A)$$ has a complement inside $$H^2$$. However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case
In terms of the general background, one way of putting this is that the skew map:

$$H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A)$$

has a section (i.e., a reverse map):

$$\operatorname{Hom}(\bigwedge^2G,A) \to H^2(G;A)$$

whose image is $$H^2_{CP}(G;A)$$ of cohomology classes represented by cyclicity-preserving 2-cocycles (see cyclicity-preserving subgroup of second cohomology group for trivial group action). Thus, the natural short exact sequence splits, and we get an internal direct sum decomposition:

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

A pictorial description of this is 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.

Generalized Baer Lie rings
The direct sum decomposition (discussed in the preceding section):

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

gives rise to some examples of the cocycle halving generalization of Baer correspondence. For any group extension arising as an element of $$H^2(G;A)$$, the additive group of its Lie ring arises as the group extension corresponding to the projection onto $$H^2_{sym}(G;A)$$, and the Lie bracket coincides with the group commutator.

In the description below, the additive group of the Lie ring of a given group is the unique abelian group in the column corresponding to that group.

Thus, we have two correspondences emerging: