  Let $A$ be a finite nilpotent group acting fixed point freely on the finite
(solvable) group $G$ by automorphisms. It is conjectured that the nilpotent
length of $G$ is bounded above by $\ell(A)$, the number of primes dividing the
order of $A$ counted with multiplicities. In the present paper we consider the
case $A$ is cyclic and obtain that the nilpotent length of $G$ is at most
$2\ell(A)$ if $|G|$ is odd. More generally we prove that the nilpotent length
of $G$ is at most $2\ell(A)+ \mathbf{c}(G;A)$ when $G$ is of odd order and $A$
normalizes a Sylow system of $G$ where $\mathbf{c}(G;A)$ denotes the number of
trivial $A$-modules appearing in an $A$-composition series of $G$.
