On permuteral subgroups in finite groups

The permutizer of a subgroup $H$ in a group $G$ is defined as the subgroup generated by all cyclic subgroups of $G$ that permute with $H$. Call $H$ permuteral in $G$ if the permutizer of $H$ in $G$ coincides with $G$; $H$ is called strongly permuteral in $G$ if the permutizer of $H$ in $U$ coincides with $U$ for every subgroup $U$ of $G$ containing $H$. We study the finite groups with given systems of permuteral and strongly permuteral subgroups and find some new characterizations of w-supersoluble and supersoluble groups.