The largest automorphism group of del Pezzo surface of degree $2$ without points

We construct an example of a field and a del Pezzo surface of degree $2$ over this field without points such that its automorphism group is isomorphic to ${\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_7)\times \mathbb{Z}\mathrm{/}2\mathbb{Z},$ which is the largest possible automorphism group of del Pezzo surface of degree $2$ over an algebraically closed field of characteristic zero.