$\mathscr M_p$-supplemented subgroups of finite groups

A subgroup $K$ of $G$ is $\mathscr M_p$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G=KB$ and $TB<G$ for every maximal subgroup $T$ of $K$ with $|K:T|=p^\alpha$. In this paper we prove the following: Let $p$ be a prime divisor of $|G|$ and let $H$ be a $p$-nilpotent subgroup having a Sylow $p$-subgroup of $G$. Suppose that $H$ has a subgroup $D$ with $D_p\ne1$ and $|H:D|=p^\alpha$. Then $G$ is $p$-nilpotent if and only if every subgroup $T$ of $H$ with $|T|=|D|$ is $\mathscr M_p$-supplemented in $G$ and $N_G(T_p)/C_G(T_p)$ is a $p$-group.