Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order pk with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γk−2(CG(a)) is nilpotent of class at most c for any a ∈ A#, then γk−2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2d + 2 ≤ k, the dth derived group of CG (a) is nilpotent of class at most c for any a ∈ A#, then the dth derived group G(d) is nilpotent and has {c, k, p}-bounded nilpotency class.