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.