  Let $X$ be a minimal del Pezzo surface of degree $2$ over a finite field
$\mathbb{F}_q$. The image $\Gamma$ of the Galois group
$\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group
$\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of
the Weyl group $W(E_7)$. There are $60$ conjugacy classes of cyclic subgroups
in $W(E_7)$ and $18$ of them correspond to minimal del Pezzo surfaces. In this
paper we study which possibilities of these subgroups for minimal del Pezzo
surfaces of degree $2$ can be achieved for given $q$.
