Structure of finite groups, in which any pronormal subgroup is either normal or abnormal

Abstract

A subgroup $$$H$$$ of a group $$$G$$$ is said to be abnormal in $$$G$$$ if, for each element $$$g \in G$$$, we have $$$g \in {<}H, H^g{>}$$$. A subgroup $$$H$$$ of a group $$$G$$$ is said to be pronormal in $$$G$$$ if, for each element $$$g \in G$$$, the subgroups $$$H$$$ and $$$H^g$$$ are conjugate in $$${<}H, H^g{>}$$$. We describe all finite groups, each pronormal subgroup in which is either normal or abnormal.