On $S$-quasinormally embedded subgroups of finite groups

Let $G$ be a finite group. A subgroup $A$ is called: 1) $S$-quasinormal in $G$ if $A$ is permutable with all Sylow subgroups in $G$ 2) $S$-quasinormally embedded in $G$ if every Sylow subgroup of $A$ is a Sylow subgroup of some $S$-quasinormal subgroup of $G$. Let $B_{seG}$ be the subgroup generated by all the subgroups of $B$ which are $S$-quasinormally embedded in $G$. A subgroup $B$ is called $SE$-supplemented in $G$ if there exists a subgroup $T$ such that $G=BT$ and $B\cap T\le B_{seG}$. The main result of the paper is the following.
Theorem. Let $H$ be a normal subgroup in $G$, and $p$ a prime divisor of $|H|$ such that $(p-1,|H|)=1$. Let $P$ be a Sylow $p$-subgroup in $H$. Assume that all maximal subgroups in $P$ are $SE$-supplemented in $G$. Then $H$ is $p$-nilpotent and all its $G$-chief $p$-factors are cyclic.