On the permutability of $n$-maximal subgroups with Schmidt subgroups

A Schmidt group is a nonnilpotent group in which every proper subgroup is nilpotent. Let us fix a positive integer $n$ and assume that each $n$-maximal subgroup of a finite group $G$ is permutable with any Schmidt subgroup. We prove that, if $n\in\{1,2,3\}$, then $G$ is metanilpotent and, if $n\ge4$ and $G$ is solvable, then the nilpotent length of $G$ is at most $n-1$.