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Let $G$ be a finite group with a unique involution $u$ and let $Q = G/\langle u \rangle$. I am looking for sufficient conditions on $Q$ under which the isomorphism type of $G$ is uniquely determined by that of $Q$.

The motivating examples for this question are the finite rotation groups $Q \leq SO(3)$ which can be pulled back to a finite subgroup $G$ of the unit quaternions such that $G \to Q$ is 2:1 if $|Q|$ is even. In this case the isomorphism type of $G$ is determined by that of $Q$, which I want to prove in a preferably general way. Note that $Q$ is either cyclic, dihedral, or one of $A_4$, $S_4$ or $A_5$ here.

I see that this question has a cohomological flavor since we are talking about certain central extensions of $Q$ by $C_2$ which can be investigated by looking at $H^2(Q,C_2)$. But since I have just a tiny understanding of the cohomology theory of groups, I hope that such arguments can be broken down to a quite elementary level.

I would be happy about any idea or hint. Thank you in advance!

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    The fact that $G$ has a unique involution means that a Sylow $2$-subgroup of $G$ is cyclic or generalized quaternion. I think that the isomorphism type of $G$ is always determined by that of $Q$. That's certainly the case when it is cyclic. Sorry but I don't have time to work out the details now and it would indeed involve consideration of $H^2(Q,C_2)$, which is isomorphic to a subgroup of $H^2(P,C_2)$ for a Sylow $2$-subgroup $P$ of $Q$. Note that $P$ is cyclic or dihedral. – 2017-01-05
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    @DerekHolt: It is okay for me to consider $H^2$ in principle - at least I know the explicit construction via 2-cocycles, and its correspondence to group extensions. So I see there is a natural homomorphism $H^2(Q,C_2) \to H^2(P,C_2)$. Do you have a reference, why it is injective? – 2017-01-05

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