Invariant TI-subgroups and structure of finite groups

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