19.109 (2018)

A subgroup $H$ of a finite group $G$ is called pronormal if for any $g \in G$ the subgroups $H$ and $H^g$ are conjugate by an element of $\langle H, H^g \rangle$. A maximal subgroup of a maximal subgroup is called second maximal. Is it true that in a non-abelian finite simple group $G$ all maximal subgroups are Hall subgroups if and only if every second maximal subgroup of $G$ is pronormal in $G$?