Extensions for trivial outer action of Z2 on SL(2,3)

We consider here the group extensions where the base normal subgroup $$N$$ is special linear group:SL(2,3) (order 24), the quotient group $$Q$$ is cyclic group:Z2 (order 2), 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 Z2 on Z2, which is isomorphic to cyclic group:Z2.