On weakly $\mathrm S$-embedded and weakly $\tau$-embedded subgroups

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be weakly $\mathrm S$-embedded in $G$ if there exists a normal subgroup $K$ of $G$ such that $HK$ is $\mathrm S$-quasinormal in $G$ and $H\cap K\le H_{seG}$, where $H_{seG}$ is the subgroup generated by all those subgroups of $H$ which are $\mathrm S$-quasinormally embedded in $G$. We say that a subgroup $H$ of $G$ is weakly $\tau$-embedded in $G$ if there exists a normal subgroup $K$ of $G$ such that $HK$ is $\mathrm S$-quasinormal in $G$ and $H\cap K\le H_{\tau G}$, where $H_{\tau G}$ is the subgroup generated by all those subgroups of $H$ which are $\tau$-quasinormal in $G$. In this paper, we study the properties of weakly $\mathrm S$-embedded and weakly $\tau$-embedded subgroups, and use them to determine the structure of finite groups.