The influence of weakly $S\Phi$-supplemented subgroups on the structure of finite groups

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be $s$-permutable in $G$ if $HP=PH$ for all Sylow subgroups $P$ of $G$. A subgroup $H$ of $G$ is said to be weakly $S\Phi$-supplemented in $G$ if $G$ has a subgroup $K$ such that $G=HK$ and $H\cap K\leq \Phi (H)H_{sG}$, where $\Phi(H)$ is the Frattini subgroup of $H$, and $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-permutable in $G$. In this paper, we investigate the structure of $G$ under the assumption that certain subgroups of fixed prime power orders are weakly $S\Phi$-supplemented in $G$. Our main results improve and extend some results in the literature.