On weakly $S\Phi$-supplemented subgroups of finite groups

Let $G$ be a finite group. We say that a subgroup $H$ of $G$ is weakly $S\Phi$-supplemented in $G$ if $G$ has a subgroup $T$ such that $G=HT$ and $H\cap T\le\Phi(H)H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ that are $s$-permutable in $G$. In this paper, we investigate the influence of weakly $S\Phi$-supplemented subgroups on the structure of finite groups. Some new characterizations of $p$-nilpotency and supersolubility of finite groups are obtained.