Invariant TI-subgroups and structure of finite groups

Let G be a finite group and assume that a group of automorphisms A is acting on G such that A and G have coprime orders. Recall that a subgroup H of G is said to be a TI-subgroup if it has trivial intersection with its distinct conjugates in G. We study the solubility and other properties of G when we assume that certain invariant subgroups of G are TI-subgroups, precisely when all A-invariant subgroups, all non-nilpotent A-invariant subgroups, and all non-abelian A-invariant subgroups of G, respectively, are TI-subgroups. (C) 2020 Elsevier B.V. All rights reserved.