Implications on the Index of a Fixed Point Subgroup 2

Let G be a finite group and A be a subgroup of Aut(G). In this work, we study the influence of the index of the fixed-point subgroup of A in G on the structure of G. When A is cyclic, we prove that G is solvable if the index of the centralizer of each x in H-G is square-free, where H denotes the semidirect product of G by A.