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 xinX we have AL^x=L^xA. We say that H is emph{X-quasipermutable } (emph{XS-quasipermutable}, respectively) in G provided G has a subgroup B such that G=NG(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 XS-quasipermutable subgroups on the structure of G. Some known results are generalized.