20.83 (2022)

Let $\mathfrak{X}$ be a class of finite groups that is closed with respect to taking subgroups, homomorphic images, and extensions. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{X}$-submaximal if there exists an embedding of $G$ into a group $G^*$ such that $G$ is subnormal in $G^*$ and $H$ coincides with the intersection of $G$ and an $\mathfrak{X}$-maximal subgroup of $G^*$.

Suppose that all $\mathfrak{X}$-submaximal subgroups of a characteristic subgroup $N$ of a finite group $G$ are conjugate in $N$. Does it follow that $HN/N$ is an $\mathfrak{X}$-submaximal subgroup of $G/N$ for every $\mathfrak{X}$-submaximal subgroup $H$ of $G$?

For normal subgroups $N$, this is not true even if $N$ is an $\mathfrak{X}$-group or if $N$ does not contain nontrivial $\mathfrak{X}$-subgroups. A positive answer is known in the case where $N$ coincides with the $\mathfrak{F}$-radical of $G$ for a Fitting class $\mathfrak{F}$.