Extensions for trivial outer action of V4 on Q8

We consider here the group extensions where the base normal subgroup $$N$$ is quaternion group, the quotient group $$Q$$ is Klein four-group, and the induced outer action of the quotient group on the normal subgroup is trivial.

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

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

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 extensions for the trivial outer action of $$Q$$ on $$N$$ correspond to the elements of the second cohomology group for trivial group action:

$$\! H^2(Q;Z(N))$$

The correspondence is as follows: an element of $$H^2(Q;Z(N))$$ gives an extension with base $$Z(N)$$ and quotient $$Q$$. We take the central product of this extension group with $$N$$, identifying the common $$Z(N)$$.

See second cohomology group for trivial group action of V4 on Z2, which is isomorphic to elementary abelian group:E8.

Extensions
The table below builds on the knowledge of second cohomology group for trivial group action of V4 on Z2.