Automorphisms of finite groups of bounded rank

LetG be a finite group admitting an automorphismα withm fixed points. Suppose every subgroup ofG isr-generated. It is shown that (1)G has a characteristic soluble subgroupH whose index is bounded in terms ofm andr, and (2) if the orders ofα andG are coprime, then the derived length ofH is also bounded in terms ofm andr. To Professor John Thompson, in honor of his outstanding achievements