A classification of degree $2$ semi-stable rational maps ${\mathbb{P}}^2\to {\mathbb{P}}^2$ with large finite dynamical automorphism group

Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the ${\text{PGL}}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\mathbb{P}}_K^2⇢{\mathbb{P}}_K^2$ whose automorphism group $$\text{Aut}(f):=\{\varphi \in {\text{PGL}}_3(K):{\varphi }^{-1}\circ f\circ \varphi =f\}$$ is finite and of order at least $3$. In particular, we prove that $\mathrm{\#}\text{Aut}(f)\le 24$ in general, that $\mathrm{\#}\text{Aut}(f)\le 21$ for morphisms, and that $\mathrm{\#}\text{Aut}(f)\le 6$ for all but finitely many conjugacy classes of $f$.