The Largest Automorphism Group of a Del Pezzo Surface of Degree $$2$$ without Points

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