  Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$
denote the characteristic polynomial of the isometry $f^*$ of the numerical
N\'eron-Severi lattice of $Y$ induced by $f$. We apply a modification of
McMullen's method to prove that the modulo-$2$ reduction $(p_f(x) \bmod 2)$ is
a product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials
$\Phi_m$, where $m \leq 9$ and $m$ is odd. We study Enriques surfaces that
realize modulo-$2$ reductions of $\Phi_7$, $\Phi_9$ and show that each of the
five polynomials $(\Phi_m(x) \bmod 2)$ is a factor of the modulo-$2$ reduction
$(p_f(x) \bmod 2)$ for a complex Enriques surface.
