On $p$-supersolubility of one class finite groups

The following is proved: A finite group $G$ is $p$-supersoluble if and only if it has a normal subgroup $N$ with $p$-supersoluble quotient $G / N$ such that either $N$ is $p'$-group or $p$ divides $|N|$ and $|G : N_G(L)|$ equals to a power of $p$ for any cyclic $p$-subgroup $L$ of $N$ of order $p$ or order $4$ (if $p = 2$ and a Sylow $2$-subgroup of $N$ is non-abelian).