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The influence of $\mathfrak{F_{\mathrm s}}$-quasinormality of subgroups on the structure of finite groups

Let $\frak{F}$ be a class of finite groups. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{F_{\mathrm s}}$-quasinormal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-permutable in $G$ and $(H\cap T)H_G/H_G$ is contained in the $\frak{F}$-hypercenter $Z_\infty^\mathfrak{F}(G/H_G)$ of $G/H_G$. In this paper, we investigate further the influence of $\mathfrak{F_{\mathrm s}}$-quasinormality of some subgroups on the structure of finite groups. New characterization of some classes of finite groups are obtained.