p-Nilpotent maximal subgroups in finite groups

Let $p$ be a prime number and suppose that every maximal subgroup of a finite group is either $p$-nilpotent or has prime index. Such group need not be $p$-solvable, and we study its structure by proving that only one nonabelian simple group of order divisible by $p$, which belongs to the family ${\rm PSL}_n(q)$, can be involved in it. For $p=2$, we specify more, and in fact, such simple group must be isomorphic to ${\rm PSL}_2({r^a})$ for certain values of the prime $r$ and the parameter $a$.