On finite groups with modular Schmidt subgroup

Let $G$ be a finite group. Then $G$ is called a Schmidt group if $G$ is not nilpotent but every proper subgroup of $G$ is nilpotent. A subgroup $M$ of $G$ is called modular in $G$ if $M$ is a modular element (in the sense of Kurosh) of the lattice $L(G)$ of all subgroups of $G$, that is, (i) $\langle X, M\cap Z \rangle=\langle X, M\rangle\cap Z$ for all $X\leqslant G$, $Z\leqslant G$ such that $X\leqslant Z$, and (ii) $\langle M, Y\cap Z \rangle=\langle M, Y\rangle\cap Z$ for all $Y\leqslant G$, $Z\leqslant G$ such that $M\leqslant G$. In this paper, we prove that if every Schmidt subgroup $A$ of $G$ with $A\leqslant G'$ is modular in $G$, then $G$ is soluble, and if every Schmidt subgroup of $G$ is modular in $G$, then the derived subgroup $G'$ is nilpotent.