On permutability of $n$-maximal subgroups with $p$-nilpotent Schmidt subgroups

A Schmidt group is a finite nonnilpotent group in which every proper subgroup is nilpotent. Fix a positive integer $n.$ Let $G$ be a solvable group. Suppose that each $n$-maximal subgroup of $G$ is permutable with every $p$-nilpotent Schmidt subgroup. We prove that if $n\in\{1,2,3\},$ then $G/F(G)$ is $p$-closed, where $F(G)$ is the Fitting subgroup of $G$.