Finite groups with weakly subnormal Schmidt subgroups

A non-nilpotent finite group whose all proper subgroups are nilpotent is called a Schmidt group. A subgroup $H$ of a group $G$ is called weakly subnormal in $G$ if $H$ is generated by two subgroups, one of which is subnormal in $G$ and the other is seminormal in $G$. We establish $3$-solvability of a finite group with weakly subnormal $\{2,3\}$-Schmidt subgroups. This implies solvability of a finite group with weakly subnormal $\{2,3\}$-Schmidt subgroups and $5$-closed $\{2,5\}$-Schmidt subgroups. We prove nilpotency of the derived subgroup of a finite group in which all Schmidt subgroups are weakly subnormal.