  In this paper we find the exact value of the Jordan constant for Cremona
group of rank $2$ over all finite fields. During the proof we construct a cubic
surface over $\mathbb{F}_2$ with a regular action of the group $\mathrm{S}_6$
which is the maximal automorphism group of cubic surfaces over $\mathbb{F}_2.$
Moreover, we prove the uniqueness up to isomorphism of such a cubic surface.
