Extensions for trivial outer action of Z2 on D16

We consider here the group extensions where the base normal subgroup $$N$$ is dihedral group:D16, the quotient group $$Q$$ is cyclic group:Z2, 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:

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

See second cohomology group for trivial group action of Z2 on Z2, which is isomorphic to cyclic group:Z2.