A condition for the solvability of finite groups

A subgroup $H$ is called $\mathscr M$-supplemented in a finite group $G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and $H_1B$ is a proper subgroup of $G$ for every maximal subgroup $H_1$ of $H$. We investigate the influence of $\mathscr M$-supplementation of Sylow subgroups and obtain a condition for solvability and $p$-supersolvability of finite groups.