17.45 (2010)

A subgroup $H$ of a group $G$ is called pronormal if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. We say that $H$ is strongly pronormal if $L^g$ is conjugate to a subgroup of $H$ in $\langle H, L^g \rangle$ for every $L \leqslant H$ and $g \in G$.
$\qquad$ a) In a finite simple group, are Hall subgroups always pronormal?
$\qquad$ b) In a finite simple group, are Hall subgroups always strongly pronormal?
$\qquad$ c) In a finite group, is a Hall subgroup with a Sylow tower always strongly pronormal?

Notice that there exist finite (non-simple) groups with a non-pronormal Hall subgroup. Hall subgroups with a Sylow tower are known to be pronormal.