On maximal subgroups of finite groups

In 1986 V.A. Vedernikov proved that if $M$ is a non-normal maximal subgroup of a finite soluble group $G$, then $M$ contains a normalizer of some Sylow subgroup of $G$. In the paper the following generalization of Vedernikov’s result is proved.

Theorem. Let $G$ be a $\pi$-soluble finite group. Let $M$ be a non-normal maximal subgroup of $G$ such that $|G : M|$ is a power of a prime $p$ in $\pi$. Let H be a Hall subgroup in $M$ such that $p$ does not divide $|H|$, and either $|\pi(H) \cap \pi'|\le 1$ or $|M : H|$ is a $\pi$-number. If the core of $HM_G / M_G$ in $M / M_G$ is not equal to $1$, then $N_G(H)$ is contained in $M$.

Here $M_G$ is the core of $M$ in $G$, i. e., the largest normal subgroup in $G$ contained in $M$; $\pi(H)$ is the set of prime divisors of $|H|$.