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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a \in A^{\#}}$$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a \in A^{\#}}$$\end{document}, then the dth derived group G(d) is nilpotent and has {c, k, p}-bounded nilpotency class.