Extensions for nontrivial outer action of Z4 on Q8

We consider here the group extensions where the base normal subgroup $$N$$ is dihedral group:D8, the quotient group $$Q$$ is cyclic group:Z4, and the induced outer action of the quotient group on the normal subgroup is a nontrivial map. There are three such possible maps, but they are all conjugate to each other, and hence there is essentially only one type of map.

More explicitly, note that $$\operatorname{Out}(N)$$ is isomorphic to symmetric group:S3, with three conjugate copies of cyclic group:Z2 in it (the three S2 in S3s), and thus there is a unique nontrivial map from $$Q$$ to cyclic group:Z2.

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 Z2 on Z2, and is isomorphic to cyclic group:Z2. The extension set is thus a set of size two with this group acting on it.