= Group extension problem
{wiki=Group_extension}
Besides the understandable Wikipedia definition, <video Simple Groups - Abstract Algebra by Socratica (2018)> gives an understandable one:
\Q[Given a finite group $F$ and a simple group $S$, find all groups $G$ such that $N$ is a <normal subgroup> of $G$ and $G/N = S$.]
We don't really know how to make up larger groups from smaller simple groups, which would complete the <classification of finite groups>:
* https://math.stackexchange.com/questions/25315/how-is-a-group-made-up-of-simple-groups
In particular, this is hard because you can't just take the <direct product of groups> to retrieve the original group: <relationship between the quotient group and direct products>{full}.