Finite Groups with Some Maximal Subgroups of Sylow Subgroups $\mathscr M$-Supplemented

A subgroup $H$ of a group $G$ is said to be $\mathscr M$‑supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G=HB$ and $TB<G$ for every maximal subgroup $T$ of $H$. In this paper, we obtain the following statement: Let $\mathscr F$ be a saturated formation containing all supersolvable groups and $H$ be a normal subgroup of $G$ such that $G/H\in\mathscr F$. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of $F^{*}(H)$, having no supersolvable supplement in $G$, is $\mathscr M$-supplemented in $G$. Then $G\in\mathscr F$.