Noncoprime action of a cyclic group

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 $\mathrm{\ell }(A)$, 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 $2\mathrm{\ell }(A)$ if $\mathrm{\mid }G\mathrm{\mid }$ is odd. More generally we prove that the nilpotent length of $G$ is at most $2\mathrm{\ell }(A)+\mathbf{c}(G;A)$ 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 $G$.