Finite groups with weakly $S\Phi$-supplemented minimal subgroups

Let $G$ be a finite group. A subgroup $H$ of $G$ is called weakly $S \Phi$-supplemented in $G$ if $G$ has a subgroup $T$ such that $G=H T$ and $H \cap T \leq \Phi(H) H_{s G}$, where $H_{sG}$ is the subgroup generated by all those subgroups of $H$ that are $s$-permutable in $G$. We investigate the structure of finite groups in which certain minimal subgroups satisfy the weakly $S \Phi$-supplemented property, obtainsome new characterizations of nilpotent groups and improve several recent theorems.