Let $G$ be a finite group and let $M$ be a $G$-module that is a finite abelian $p$-group. Suppose we have extensions

$1 \rightarrow M \rightarrow E_1 \rightarrow G \rightarrow 1$

and

$1 \rightarrow M \rightarrow E_2 \rightarrow G \rightarrow 1$

such that the extensions have a common restriction

$1 \rightarrow M \rightarrow F \rightarrow H \rightarrow 1$

where $F \le E_1 \cap E_2$ such that the index of $F$ in $E_1$ and $E_2$ is coprime to $p$. Then $H$ has index coprime to $p$ in $G$, so the restriction map $\mathrm{H}^2(G,M) \rightarrow \mathrm{H}^2(H,M)$ is injective, and both extensions of $G$ by $M$ give rise to the same element of $\mathrm{H}^2(H,M)$ (corresponding to the equivalence class of $F$), hence they must also give rise to the same element of $\mathrm{H}^2(G,M)$. In other words there is an equivalence of extensions $\theta:E_1 \rightarrow E_2$.

Can we choose $\theta$ so that $\theta(x) = x$ for all $x \in F$?