$S$-embedded subgroups of finite groups

A subgroup $H$ of $G$ is said to be $S$-embedded in $G$ if $G$ has a normal subgroup $N$ such that $HN$ is $s$-permutable in $G$ and $H\cap N\le H_{sG}$, where $H_{sG}$ is the largest $s$-permutable subgroup of $G$ contained in $H$. $S$-embedded subgroups are used to give novel characterizations for some classes of groups. New results are obtained and a number of previously known ones are generalized.