On $S\Phi$-embedded subgroups of finite groups

A subgroup $H$ of $G$ is called $S\Phi$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $S$-quasinormal in $G$ and $(H \cap T)H_{G}/H_{G}\leq\Phi(H/H_{G})$, where $H_{G}$ is the maximal normal subgroup of $G$ contained in $H$ and $\Phi(H/H_{G})$ is the Frattini subgroup of $H/H_{G}$. In this paper, we investigate the influence of $S\Phi$-embedded subgroups on the structure of finite groups. In particular, some new characterizations of $p$-supersolvability of finite groups are obtained by assuming some subgroups are $S\Phi$-embedded.