Finite groups with given systems of propermutable subgroups

Let $H$ be a subgroup of a finite group $G$. Then we say that $H$ is propermutable in $G$ provided $G$ has a subgroup $B$ such that $G = NG(H)B$ and $H$ permutes with all subgroups of $B$. In this paper, we present new properties of propermutable subgroups. Also we provide new information on the structure of a group with propermutable Sylow (Hall, maximal) subgroups and a group $G = AB$ with propermutable subgroups $A$ and $B$.