Del Pezzo surfaces of degree $5$ over perfect fields

Abstract: We classify del Pezzo surfaces of degree $5$ over a perfect field $\mathbf{k}$ according to the actions of the Galois group $$\operatorname{Gal}(\overline{\mathbf{k}}/\mathbf{k})$$ on the graph of $(-1)$-curves, and we explicitely describe their automorphism groups.