Classification of finite groups satisfying a minimal condition

If $H$ is a subgroup of a finite group $G$ then we denote the normal closure of $H$ in $G$ by $H^G$. We call $G$ a $PE$-group if every minimal subgroup $X$ of $G$ satisfies $N_G(X)\cap X^G=X$. The authors classify the finite non-$PE$-groups whose maximal subgroups of even order are $PE$-groups.