On composition factors of a finite group with $OS$-seminormal Sylow subgroup

A finite non-nilpotent group whose all proper subgroups nilpotent, is called the Schmidt group. A subgroup $A$ of a group $G$ is called $OS$-seminormal, if there exists a subgroup $B$ such that $G=AB$ and $A$ commutes with all Schmidt subgroups of $B$. For a prime number $r\ge 7$ is established $r$-solvability of the group, in which the Sylow $r$-subgroup $OS$-seminormal. For $r<7$, all non-Abelian compositional factors are listed such group. The solvability of the group with $OS$-seminormal Sylow $2$- and $3$-subgroups.