On groups satisfying a positive law in fixed points

Let A be an elementary abelian q-group of order q2 acting on a finite q′-group G in such a manner that the subgroup 〈CG(a),CG(b)〉 satisfies a positive law of degree n for any a,b∈A#. It is proved that the entire group G satisfies a positive law of degree bounded by a function of q and n only.