Finite groups with $X$-quasipermutable subgroups of prime power order

‎Let $H$‎, ‎$L$ and $X$ be subgroups of a finite group‎ ‎$G$‎. ‎Then $H$ is said to be $X$-permutable with $L$ if for some‎ ‎$x\in X$ we have $AL^{x}=L^{x}A$‎. ‎We say that $H$ is‎ ‎\emph{$X$-quasipermutable } (\emph{$X_{S}$-quasipermutable}‎, ‎respectively)‎ ‎in $G$ provided $G$ has a subgroup‎ ‎$B$ such that $G=N_{G}(H)B$ and $H$ $X$-permutes‎ ‎with $B$ and with all subgroups (with all Sylow‎ ‎subgroups‎, ‎respectively) $V$ of $B$ such that $(|H|‎, ‎|V|)=1$‎. ‎In‎ ‎this paper‎, ‎we analyze the influence of $X$-quasipermutable and‎ ‎$X_{S}$-quasipermutable subgroups on the structure of $G$‎. ‎Some known‎ ‎results are generalized.