Criteria of $p$-supersolubility of finite groups

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is $\tau$-quasinormal in $G$ if $H$ permutes with all Sylow subgroups $Q$ of $G$ such that $(|Q|, |H|)=1$ and $(|H|, |Q^G|)\ne1$. The main result here is the following: Let $G=AT$, where $A$ is a Hall $\pi$-subgroup of $G$ and $T$ is $p$-nilpotent for some prime $p\notin\pi$, let $P$ denote a Sylow $p$-subgroup of $T$ and assume that $A$ is $\tau$-quasinormal in $G$. Suppose that there is a number $p^k$ such that $1<p^k<|P|$ and $A$ permutes with every subgroup of $P$ of order $p^k$ and with every cyclic subgroup of $P$ of order $4$ (if $p^k=2$ and $P$ is non-abelian). Then $G$ is $p$-supersoluble.