A stem extension of a group $X$ is a group $G$ with a subgroup $N$ contained in $G' \cap Z(G)$ such that $G/N$ is isomorphic to $X$, so that we have a short exact sequence $1 \to N \to G \to X \to 1$.
A double cover of $X$ is a stem extension $1 \to N \to G \to X \to 1$ such that $|N|=2$.
My question is: if $1 \to N \to G \to X \to 1$ is any double cover of $X$ is it always the case that the (unique) generator of the cyclic group $N$ is a square in $G$? That is, writing $N=\langle z \rangle = \{1,z\}$ is there some $x \in G$ with $x^2 = z$?
I'm talking about the general case ($X$ not necessarily perfect). It would be nice to at least get, if not an answer, a feeling of how reasonable this question is. If you have a reference about the general concept of double cover defined above and its properties it would be greatly appreciated.
Thanks!
