Jordan constant for Cremona group of rank 2 over a finite field

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 ${\mathbb{F}}_2\mathrm{.}$ Moreover, we prove the uniqueness up to isomorphism of such a cubic surface.