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Can a non-split extension of one non-Abelian finite simple group by another exist?

Let $G$ be a (profinite) group, and let $M$ be a non-abelian $G$-module. We know how to construct reasonably $H^0(G,M)$ and $H^1(G,M)$ and it turns out that $H^1(G,M)$ is just a pointed set and not ...

Let $G$ be a group and $$0\rightarrow K\rightarrow M\rightarrow N\rightarrow 0$$ a short exact sequence of groups. Now these are abelian groups, if I want to show that $\text{Hom}(G,M)\rightarrow \...

Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$). If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary ...