Finite groups with given $\rho$-semipermutable subgroups

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is $\rho$-semipermutable in $G$ if $H$ permutes with all Sylow subgroups $G_p$ of $G$ such that $(|H|,p)=1$ and $p\mid |H^{G}|$. The main purpose of this paper is to study the $p$-nilpotency of finite groups $G$ under the condition that all maximal subgroups of a Sylow $p$-subgroup of $G$ are $\rho$-semipermutable.