  Let $\mathcal{M}_{\pi}$ be an $F$-cyclic $F$-crystal $\mathcal{M}_{\pi}$ over
an algebraically closed field defined by a permutation $\pi$ and a set of
prescribed Hodge slopes. We prove combinatorial formulas for the dimension
$\gamma_{\mathcal{M}_{\pi}}(m)$ of the automorphism group scheme of
$\mathcal{M}_{\pi}$ at finite level $m$ and the number of connected components
of the endomorphism group scheme of $\mathcal{M}_{\pi}$ at finite level $m$. As
an application, we show that if $\mathcal{M}_{\pi}$ is a nonordinary
Dieudonn\'e module defined by a cycle $\pi$, then
$\gamma_{\mathcal{M}_{\pi}}(m+1) - \gamma_{\mathcal{M}_{\pi}}(m) <
\gamma_{\mathcal{M}_{\pi}}(m) - \gamma_{\mathcal{M}_{\pi}}(m-1)$ for all $1
\leq m \leq n_{\mathcal{M}_{\pi}}$, where $n_{\mathcal{M}_{\pi}}$ is the
isomorphism number of $\mathcal{M}_{\pi}$.
