Determine all indices for which the maximal-subgroup property holds

Determine the complete set of positive integers n for which every maximal subgroup of a maximal subgroup M of index n in a finite group G is a second maximal subgroup of G; equivalently, determine all n for which the assertion (\mathcal{P}_n) holds.

Background

For a positive integer n, the assertion (\mathcal{P}_n) states that, for every finite group G with a maximal subgroup M of index n, every maximal subgroup of M is a second maximal subgroup of G. The paper proves (\mathcal{P}_n) for every prime n and for every odd integer n in the set \mathcal{A} of degrees for which the only primitive subgroups of S_n are S_n and A_n.

The paper also establishes that (\mathcal{P}_n) fails for every composite even integer and every composite prime power, and gives further examples of both positive and negative cases among odd composite integers. However, these results do not determine the full set of indices n for which (\mathcal{P}_n) holds, leaving the general classification unresolved.