a question about fixed-point-free automorphism group 2

Question

In this paper, Rowley (1995), there is a theorem:

Let $A$ and $G$ be finite groups. Suppose that $A$ acts fixed-point-freely on $G$ and that either $A$ is cyclic or $(|G|,|A|)=1$. Then $G$ is soluble.

I have two questions

1. Is there a proof without recourse to the simple group classification for this theorem ?
2. $G$ is soluble, can the derived length of $G$ be bounded?

I can't communicate with the author, I will be very glad if someone can reply.

• Rowley, Peter. “Finite groups admitting a fixed-point-free automorphism group.” J. Algebra 174 (1995), no. 2, 724–727. MR1334233