Forms of del Pezzo surfaces of degree $5$ and $6$

We obtain necessary and sufficient condition for the existence of del Pezzo surfaces of degrees $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on the graph of $(-1)$-curves. We also compute the automorphism groups of del Pezzo surfaces of degree $5$ over arbitrary fields.
Bibliography: 19 titles.