The existence of pronormal $\pi$-Hall subgroups in $E_\pi$-groups

A subgroup $H$ of a group $G$ is called pronormal, if the subgroups $H$ and $H^g$ are conjugate in $\langle H,H^g\rangle$ for every $g\in G$. It is proven that if a finite group$G$ possesses a $\pi$-Hall subgroup for a set of primes $\pi$, then its every normal subgroup (in particular, $G$ itself) has a $\pi$-Hall subgroup pronormal in $G$.