Quotients of del Pezzo surfaces

Any unirational surface is rational over an algebraic closed field. But for nonclosed fields it is not true. For example, any Del Pezzo surface of degre 2, 3 or 4 is unirational but it can be nonrational. One of the important classes of unirational surfaces are quoients of rational surfaces. We will proof the following results about quotients of rational surfaces:
1) If $X$ is a Del Pezzo surface, $X(\Bbbk) \neq \varnothing$, $K_X^2 \geq 5$, $G \subset \mathrm{Aut}(X)$, then $X / G$ — $\Bbbk$-rational.
2) If $X$ is a Del Pezzo surface, $X(\Bbbk) \neq \varnothing$, $K_X^2 = 4$, $G \subset \mathrm{Aut}(X)$, $G \neq \{1\}$, then $X / G$ — $\Bbbk$-rational.
3 If $X$ is a Del Pezzo surface, $X(\Bbbk) \neq \varnothing$, $K_X^2 = 3$, $G \subset \mathrm{Aut}(X)$, $G \neq \{1\}$, $C_3$, then $X / G$ — $\Bbbk$-rational.
4) If $X$ is a Del Pezzo surface, $X(\Bbbk)$ is dense, $K_X^2 =2$, $G \subset \mathrm{Aut}(X)$, $G \neq \{1\}$, $C_2$, $C_2^2$, $C_4$, $D_4$, $Q_8$, then $X / G$ — $\Bbbk$-rational.
5) If $X$ is a Del Pezzo surface, $X(\Bbbk)$ is dense, $K_X^2 =1$, $G \subset \mathrm{Aut}(X)$, $G \neq \{1\}$, $C_2$, $C_3$, $C_6$, $S_3$, then $X / G$ — $\Bbbk$-rational.