Characterization of finite groups with some $S$-quasinormal subgroups of fixed order

Let $G$ be a finite group. A subgroup of $G$ is said to be $S$-quasinormal in $G$ if it permutes with every Sylow subgroup of $G$. We fix in every non-cyclic Sylow subgroup $P$ of the generalized Fitting subgroup a subgroup $D$ such that $1 < |D| < |P|$ and characterize $G$ under the assumption that all subgroups $H$ of $P$ with $|H| = |D|$ are $S$-quasinormal in $G$. Some recent results are generalized.