Extensions for nontrivial outer action of V4 on D8

We consider here the group extensions where the base normal subgroup $$N$$ is dihedral group:D8, the quotient group $$Q$$ is Klein four-group $$V_4$$, and the induced outer action of the quotient group on the normal subgroup is a nontrivial map to $$\operatorname{Out}(N)$$ which is isomorphic to $$\mathbb{Z}_2$$. There are three possibilities for the nontrivial map, each with kernel one of the copies of Z2 in V4. They are all equivalent under pre-composition by automorphisms of the Klein four-group.

Description in terms of cohomology groups
We have the induced outer action which is nontrivial:

$$Q \to \operatorname{Out}(N)$$

Composing with the natural mapping $$\operatorname{Out}(N) \to \operatorname{Aut}(Z(N))$$, we get a trivial map:

$$Q \to \operatorname{Aut}(Z(N))$$

Thus, the number of extensions for the trivial outer action of $$Q$$ on $$N$$ equals the number of elements in the second cohomology group for trivial group action $$H^2(Q;Z(N))$$ for the trivial group action. More explicitly, $$H^2(Q;Z(N))$$ acts on the set of extensions (possibly with repetitions) in a manner that is equivalent to the regular group action. However, the extension set does not have a natural choice of extension corresponding to the identity element.

$$H^2(Q;Z(N))$$ is the second cohomology group for trivial group action of V4 on Z2, and is isomorphic to elementary abelian group:E8. The extension set is thus a set of size two with this group acting on it.

Extensions
Note that there are two different and non-pseudo-congruent extensions both giving isomorphic overall extension groups (holomorph of Z8). See series-equivalent not implies automorphic.