Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup $A$ is said to be seminormal in a group $G$ if there exists a subgroup $B$ such that $G=AB$ and $AB_1$ is a proper subgroup of $G$, for every proper subgroup $B_1$ of $B$. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt $\{2,3\}$-subgroups and all 5-closed $\{2,5\}$-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary.