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Let $\mathcal{M}_{\pi}$ be an $F$-cyclic $F$-crystal $\mathcal{M}_{\pi}$ overan algebraically closed field defined by a permutation $\pi$ and a set ofprescribed 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 componentsof the endomorphism group scheme of $\mathcal{M}_{\pi}$ at finite level $m$. Asan application, we show that if $\mathcal{M}_{\pi}$ is a nonordinaryDieudonn\'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 theisomorphism number of $\mathcal{M}_{\pi}$.

Dimensions of automorphism group schemes of finite level truncations of $F$-cyclic $F$-crystals