Let $G$ be a group, and $A$ an abelian group on which $G$ acts. Let $N\le G$ be a normal subgroup, and let $A^N$ denote the $N$-invariants of $A$. Then we have for every $n\ge 1$, we have a sequence in cohomology: $$H^n(G/N,A^N)\rightarrow H^n(G,A^N)\rightarrow H^n(G,A)$$ If $n=2$, this allows us to produce, very abstractly, an extension $F$ of $G$ by $A$ from a given extension $E$ of $A^N$ by $G/N$.
Is there a nice description of how one might construct $F$ from $E$?
The extension of $A^N$ by $G$ corresponding to the image of $E$ under the inflation map is just the subgroup of $G \times E$ consisting of those elements in which the two components map onto the same element of $G/N$.
To get the extension $F$ of $A$ by $G$, you just can use the same cocycle $G \times G \to A^N$ and regard it as a cocycle $G \times G \to A$.