Are there any references on extensions $G$ of a cyclic group $C_2$ by $2$-groups $P$ such that $1\neq a\in C_2$ is a square element in $G$?
In other words, if $G/{C_2}\cong P$, where $P$ is a 2-group, and $1\neq a\in C_2$ is a square element in $G$, then what can we see on the structure of $G$? Are there any references on these group extensions?