A note on the $\mathrm{\Pi }$-property of some subgroups of finite groups

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the $\mathrm{\Pi }$-property in $G$ if for any chief factor $L\mathrm{/}K$ of $G$, $\mathrm{\mid }G\mathrm{/}K:N_{G\mathrm{/}K}(HK\mathrm{/}K\cap L\mathrm{/}K)\mathrm{\mid }$ is a $\pi (HK\mathrm{/}K\cap L\mathrm{/}K)$-number. In this paper, we obtain some criteria for the $p$-supersolubility or $p$-nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the $\mathrm{\Pi }$-property.