Finite groups with systems of $N$-quasinormal subgroups

Throughout the article, all groups are finite and $G$ always denotes a finite group. A subgroup $A$ of a group $G$ is called quasinormal in $G$ if $AH = HA$ for all subgroups $H$ of $G$. If $A$ is a subgroup of $G$, then $A_{qG}$ is the subgroup of $A$ generated by all those subgroups of $A$ that are quasinormal in $G$. We say that the subgroup $A$ is $N$-quasinormal in $G$ ($N\geqslant G$), if for some quasinormal subgroup of $T$ of $G$, containing $A$, $N$ avoids the pair $(T, A_{qG})$, i. e. $N\cap T=N\cap A_{qG}$. Using these concepts, we give new characterizations of soluble and supersoluble finite groups.