Automorphisms of finite groups of bounded rank

Let G be a finite group admitting an automorphism α with m fixed points. Suppose every subgroup of G is r-generated. It is shown that (1)G has a characteristic soluble subgroup H whose index is bounded in terms of m and r, and (2) if the orders of α and G are coprime, then the derived length of H is also bounded in terms of m and r.