On $X$-$s$-permutable subgroups of finite groups

Let $X$ be a nonempty subset of a group $G$. A subgroup $H$ of $G$ is said to be $X$-$s$-permutable in $G$ if for every Sylow subgroup $T$ of $G$, there exists an element $x\in X$ such that $HT^x=T^xH$. In this paper we obtain some results on $X$-$s$-permutable subgroups and use them to determine the structure of some finite groups.