Finite groups with $n\Phi$-subgroups of prime orders

A subgroup $H$ of a group $G$ is called $n\Phi$-subgroup in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K\leqslant \Phi(H)$. It has been proved that any formation admits characterization by certain $n\Phi$-subgroups of prime orders.