Groups of automorphisms with TNI-centralizers

A subgroup H of a finite group G is called a TNI-subgroup if NG(H)∩Hg=1 for any g∈G\NG(H). Let A be a group acting on G by automorphisms where CG(A) is a TNI-subgroup of G. We prove that G is solvable if and only if CG(A) is solvable, and determine some bounds for the nilpotent length of G in terms of the nilpotent length of CG(A) under some additional assumptions. We also study the action of a Frobenius group FH of automorphisms on a group G if the set of fixed points CG(F) of the kernel F forms a TNI-subgroup, and obtain a bound for the nilpotent length of G in terms of the nilpotent lengths of CG(F) and CG(H).