On Nearly $S$-Permutably Embedded Subgroups of Finite Groups

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be $S$-permutable in $G$ if $HP=PH$ for all Sylow subgroups $P$ of $G$. A subgroup $A$ of a group $G$ is said to be $S$-permutably embedded in $G$ if for each Sylow subgroup of $A$ is also a Sylow of some $S$-permutable subgroup of $G$.
In this paper, we analyze the following generalization of this concept. Let $H$ be a subgroup of a group $G$. Then we say that $H$ is nearly $S$-permutably embedded in $G$ if $G$ has a subgroup $T$ and an $S$-permutably embedded subgroup $C\le H$ such that $HT=G$ and $T\cap H\le C$.
We study the structure of $G$ under the assumption that some subgroups of $G$ are nearly $S$-permutably embedded in $G$. Some known results are generalized.