Second cohomology group for trivial group action of D8 on Z4

Description of the group
We consider here the second cohomology group for trivial group action of dihedral group:D8 on cyclic group:Z4, i.e., the group

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

where $$G \cong D_8$$ is dihedral group:D8 (order 8, degree 4) and $$A \cong \mathbb{Z}_4$$ is cyclic group:Z4.

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

Elements
Note that all these extensions are central extensions with the base normal subgroup isomorphic to cyclic group:Z4 and the quotient group isomorphic to dihedral group:D8. Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each of the extension groups is $$4 \times 8 = 32$$.

None of the extensions is a stem extension, because the Schur multiplier of dihedral group:D8 is cyclic group:Z2, which does not admit cyclic group:Z4 as a quotient.

The minimum size of generating set of the extension group is at least equal to 2 (which is the minimum size of generating set of the quotient group) and at most equal to 3 (which is the sum of the minimum size of generating set of the normal subgroup and the quotient group). See minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group and minimum size of generating set of quotient group is at most minimum size of generating set of group.

The nilpotency class is at least 2 and at most 3 in all cases. It is at least 2 because the quotient dihedral group:D8 has nilpotency class two. It is at most 3 because the sum of the nilpotency class of the normal subgroup and quotient group is 3, and the extension is a central extension. The derived length is always exactly 2 because nilpotency class 2 or 3 forces derived length exactly 2, using derived length is logarithmically bounded by nilpotency class.

Under the action of the automorphism group of the acting group
By pre-composition, the automorphism group of dihedral group:D8, which is itself isomorphic to dihedral group:D8, acts on the second cohomology group. Under this action, there are four fixed points: direct product of D8 and Z4, nontrivial semidirect product of Z4 and Z8, central product of D16 and Z4, and SmallGroup(32,15). There are two orbits of size two, corresponding to the extensions SmallGroup(32,5) and wreath product of Z4 and Z2 respectively.

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

$$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0$$

where $$G^{\operatorname{ab}}$$ is the abelianization of $$G$$ and its image comprises those extensions where the restricted extension of the derived subgroup $$[G,G]$$ on $$A$$ is trivial and the corresponding extension of the quotient group is abelian. Also, $$H_2(G;\mathbb{Z})$$ is the Schur multiplier of $$G$$.

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

In this case
For this choice of $$G$$ and $$A$$, $$G^{\operatorname{ab}}$$ is isomorphic to the Klein four-group. The corresponding group $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ is also a Klein four-group.

The Schur multiplier $$H_2(G;\mathbb{Z})$$ is cyclic group:Z2, hence $$\operatorname{Hom}(H_2(G;\mathbb{Z}),A)$$ is also isomorphic to cyclic group:Z2.

The image of $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ in $$H^2(G;A)$$ comprises the four extensions for symmetric cohomology classes, and these are precisely the ones that yield groups that overall have nilpotency class two. It has two cosets in the whole second cohomology group, with its non-identity coset comprising precisely the extensions that have overall nilpotency class three.

A natural choice of complement is the subgroup that arises as the kernel of the homomorphism $$H^2(G;A) \to H^2(G;\mathbb{Z}_2)$$ induced by the natural surjective map from $$A = \mathbb{Z}_4$$ to $$\mathbb{Z}_2$$. This subgroup comprises precisely the trivial extension (direct product of D8 and Z4) and central product of D16 and Z4.

Below is a depiction of the elements of the second cohomology group, where the rows are cosets of the image of $$\operatorname{Ext}^1$$ and the columns are cosets of our chosen complement:

Homomorphisms on $$A$$
The unique injective homomorphism $$\mathbb{Z}_2$$ to $$A = \mathbb{Z}_4$$ induces a homomorphism:

$$\! H^2(G;\mathbb{Z}_2) \to H^2(G;A)$$

The group on the left is also isomorphic to elementary abelian group:E8 (see second cohomology group for trivial group action of D8 on Z2). However, the induced map above is not an isomorphism. Rather, it has kernel of order four precisely the image of $$\operatorname{Ext}^1(G^{\operatorname{ab}},\mathbb{Z}_2)$$ in $$H^2(G;\mathbb{Z}_2)$$ (see the direct sum decomposition section) and its image is a subgroup of order two in $$H^2(G;\mathbb{Z}_4)$$.

In terms of extensions, the map is interpreted as follows: it involves taking the central product of a given extension with cyclic group:Z4, identifying the base cyclic group:Z2 in the original extension with the $$\mathbb{Z}_2$$ in $$\mathbb{Z}_4$$.

The map is given in the table below:

The unique surjective map from $$A = \mathbb{Z}_4$$ to $$\mathbb{Z}_2$$ induces a homomorphism:

$$H^2(G;A) \to H^2(G;\mathbb{Z}_2)$$

The kernel of this map is the image of the preceding map and the image of this map is the kernel of the preceding map. The map is given in the table below: