  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.
