On the permutability of a Sylow subgroup with Schmidt subgroups from a supplement

A Schmidt group is a finite nonnilpotent group each of whose proper subgroups is nilpotent. A supplement of a subgroup $A$ in a group $G$ is a subgroup $B$ of $G$ such that $G=AB$. Finite groups in which a Sylow subgroup is permutable with some Schmidt subgroups were studied by Ya.G. Berkovich and E.M. Pal'chik (Sib. Mat. Zh. 8(4), 741-753 (1967)) and by V. N. Knyagina and V.S. Monakhov (Proc. Steklov Inst. Math. 272 (Suppl. 1), S55-S64 (2011)). In this situation, the group may be nonsolvable. For example, in the group PSL(2,7) a Sylow 2-subgroup is permutable with all Shmidt subgroups of odd order. In the group SL(2,8) a Sylow 3-subgroup is permutable with all 2-closed Shmidt subgroups of even order. In the group SL(2,4) a Sylow 5-subgroup is permutable with every 2-closed Shmidt subgroup of even order. Since the groups Sz$(2^{2k+1})$ for $k\geq 1$, PSU(5,4), PSU(4,2), and PSp$(4,2^n)$ do not contain Shmidt subgroups of odd order, in these groups any Sylow subgroup is permutable with any Shmidt subgroup of odd order. We establish the $r$-solvability a finite group $G$ such that $r$ is odd and is not a Fermat prime and a Sylow $r$-subgroup $R$ is permutable with 2-nilpotent (or 2-closed) Schmidt subgroups of even order from some supplement of $R$ in $G$. We give examples showing that the constraints on $r$ are not superfluous.