On finite groups with automorphisms whose fixed points are Engel

The main result of the paper is the following theorem. Let q be a prime, n a positive integer, and A an elementary abelian group of order q2. Suppose that A acts coprimely on a finite group G and assume that for each a∈A#\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} every element of CG(a) is n-Engel in G. Then the group G is k-Engel for some {n,q}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{n,q\}}$$\end{document}-bounded number k.