Let $G$ be a group which is the extension of a free abelian group $A$ of finite rank by a finite simple group $S$. Does $G$ splits over $A$? (that is, $G=F\ltimes A$ for some finite subgroup $F\simeq S$). Otherwise, could you provide me with some examples of a non-splitting such extension?
The fact is that we can always take a prime $p$ such that $p$ does not divide the order of $S$, so $G/A^p$ splits over $A/A^p$. I really feel an extension of this type should split, but I cannot manage to prove how.