Fixed points of coprime operator groups

Let m be a positive integer and A an elementary abelian group of order qr with r⩾2 acting on a finite q′-group G. We show that if for some integer d such that d2⩽r−1 the dth derived group of CG(a) has exponent dividing m for any a∈A#, then G(d) has {m,q,r}-bounded exponent and if γr−1(CG(a)) has exponent dividing m for any a∈A#, then γr−1(G) has {m,q,r}-bounded exponent.