Let $M$ be an $F$-crystal over an algebraically closed field of positive characteristic. For every integer $m \geq 1$, let $\gamma_{M}(m)$ be the dimension of the automorphism group scheme $\mathrm{Aut}_m(M)$ of $M$ at finite level $m$. In 2012, Gabber and Vasiu proved that $0 \leq \gamma_{M}(1) < \gamma_{M}(2) < \cdots < \gamma_{M}(n_{M}) = \gamma_{M}(n_{M}+1) = \cdots$ where $n_{M}$ is the isomorphism number of $M$, and that $\gamma_{M}(m+1)- \gamma_{M}(m) \leq \gamma_{M}(m)- \gamma_{M}(m-1)$ for all $m \geq 1$ if $M$ is a Dieudonné module over $k$. We generalized the same result to arbitrary $F$-crystals in 2014. Questions have been asked as to whether $\gamma_{M}(m+1)- \gamma_{M}(m) < \gamma_{M}(m)- \gamma_{M}(m-1)$ for all $1 \leq m \leq n_{M}$ for any $F$-crystal $M$. In this talk, we will discuss a combinatorial formula that calculates $\gamma_{M}(m)$ for a certain family of $F$-crystals called $F$-cyclic $F$-crystals. This formula allows us to give a negative answer to the aforementioned question in general but a positive answer for some families of Dieudonné modules.