Finite groups in which every non-nilpotent maximal invariant subgroup of order divisible by p is a TI-subgroup

Let A and G be finite groups such that A acts coprimely on G by automorphisms. We obtain a complete description of the structure of finite groups in which every non-nilpotent maximal A-invariant subgroup of order divisible by p is a TI-subgroup for any fixed prime divisor p of the order of groups.