On the solvability of a finite group with a pair of non-conjugate subgroups of primary indices II

The solvability of a finite group $G$ with two non-conjugate maximal subgroups $A$ and $B$ that satisfy the following requirements has been proved: subgroups $A$ and $B$ have primary indices in $G$; all proper subgroups of $A$ and $B$ are $2$-nilpotent. In addition, if $G$ is $S_4$-free and the indices of the subgroups $A$ and $B$ are coprime, then the $2$-nilpotency of the subgroups $A$ and $B$ can be replaced by their solvability.