Permuteral subgroups and their applications in finite groups

Let $H$ be a subgroup of a group $G$. The permutizer of $H$ in $G$ is the subgroup $P_G(H)=\langle x\in G | \langle x\rangle H=H\langle x\rangle\rangle$. The subgroup $H$ of a group $G$ is called permuteral in $G$, if $P_G(H)=G$; strongly permuteral in $G$, if $P_U(H)=U$ whenever $H\leqslant U\leqslant G$. The properties of finite groups with given systems of permuteral and strongly permuteral subgroups are obtained. New criteria of w-supersolubility and supersolubility of groups are received.