Let p be a fixed-point-free automorphism of order $p^{n}$ ($p$ odd prime) acting on the finite solvable group $G$. If $N_{(i)} = [G,\langle \rho^{p^{n-i}} \rangle]$ then $1 = N_{(0)} \le \ldots \le N_{(i-1)} \le N_{(i)}\le \ldots \le N_{(n)} = G$ is a normal chain with nilpotent factors $N_{(i)}/N_{(i-1)}$.