It is known that $\mathbb{P}^1 \times \mathbb{P}^1 \not \cong \mathbb{P}^2$. One way to see this is working with their respective class groups, $\mathbb{Z}^2$ and $\mathbb{Z}$, which in this case, since they are normal toric varieties are fairly easy to compute.
My question is whether we can "fix" this by identifying some points on $X =\mathbb{P}^1 \times \mathbb{P}^1$, that is, if there exists some free group action $\sigma \colon G \times X \to X$ ($G$ finite) such that the quotient $X/G$ is isomorphic to $\mathbb{P}^2$.
Forgetting the algebraic-geometry structure, the action of $\mathbb{Z}_2$ on $X$ via permutation of elements is isomorphic to $\mathbb{P}^2$, so:
Does this extend to an isomorphism of schemes?
Also, any references that deal with this type of quotients will be welcome, of course. Thank you.
Note: I wrote that the action is free because apparently this guarantees that the quotient is at least an algebraic space; the finiteness requirement guarantees that it is actually a scheme, as in the answers to this question.