On $\Pi$-property and $\Pi$-normality of subgroups of finite groups. II

Let $H$ be a subgroup of a group $G$. We say that $H$ satisfies the $\Pi$-property in $G$ if $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K)$-number for any chief factor $L/K$ of $G$. If there is a subnormal supplement $T$ of $H$ in $G$ such that $H\cap T\le I\le H$ for some subgroup $I$ satisfying the $\Pi$-property in $G$, then $H$ is said to be $\Pi$-normal in $G$. Using these properties that hold for some subgroups, we derive new $p$-nilpotency criteria for finite groups.