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 $f:{mathbb P}^2_Kdashrightarrow{mathbb P}^2_K$ whose automorphism group $$text{Aut}(f):={phiintext{PGL}_3(K): phi^{-1}circ fcircphi=f}$$ is finite and of order at least $3$. In particular, we prove that $#text{Aut}(f)le24$ in general, that $#text{Aut}(f)le21$ for morphisms, and that $#text{Aut}(f)le6$ for all but finitely many conjugacy classes of $f$.