Extensions of Abelian groups to non-Abelian groups

Question

Let $N$ and $M$ be two finite Abelian groups. Is there a nice way to characterize all extensions of $M$ by $N$? I have seen a few sources where Abelian extensions of Abelian groups are discussed but we are interested in all (both non-Abelian and Abelian) extensions. Any comment could be helpful!

Answer 1

Such extensions are characterised by the second group cohomology $H^2(M,N)$. For example, with $M=C_2$ and $N=C_3$ we have also extensions to nonabelian groups, i.e., $$ 1\rightarrow C_3 \rightarrow S_3\rightarrow C_2\rightarrow 1. $$ In other words, the symmetric group $S_3$ is an extension of $C_3$ by $C_2$. Of course, we also have the extension $$ 1\rightarrow C_3 \rightarrow C_6\rightarrow C_2\rightarrow 1. $$