On permutable strongly $2$-maximal and strongly $3$-maximal subgroups

We describe the structure of finite solvable non-nilpotent groups in which every two strongly $n$-maximal subgroups are permutable ($n = 2, 3$). In particular, it is shown for a solvable non-nilpotent group $G$ that any two strongly $2$-maximal subgroups are permutable if and only if $G$ is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable $3$-maximal subgroups and with permutable strongly $3$-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly $3$-maximal subgroups, and we describe $14$ classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly $n$-maximal subgroups if the number of prime divisors of the order of this group strictly exceeds $n$ ($n=2, 3$).