Can I get a map $f: \Bbb P^2 \to \Bbb P^2$ defined by the monomial dehomogenization $[z_1, z_2] → [z_1^2z_2, z_1z_2]$ birational such that $fσ=σf$?

Question

Consider the The standard quadratic involution.$ σ: \Bbb P^2 \to \Bbb P^2$ defined by the monomial dehomogenization $[z_1, z_2] → [z_1^2z_2, z_1z_2]$ which has the inverse $[z_1, z_2] → [z_1z_2^{-1}, z_1^{-1}z_2^2]$. So we can get the homogeneous map after homogenizing the corresponding map and I calculated the maps after taking $x_i=z_i/z_0$ $ σ: \Bbb P^2 \to \Bbb P^2$ would be $[x_0,x_1,x_2] \mapsto [x_0^3,x_1^2x_2,x_0x_1x_2]$ and the inverse map would be $[x_0,x_1,x_2] \mapsto [x_0x_1x_2,x_0x_1^2,x_2^3]$

1. Can I get a map $f: \Bbb P^2 \to \Bbb P^2$ birational such that $fσ=σf$?
2. What can we say about $BiRat(f)=\{g:\Bbb P^2 \to \Bbb P^2$ birational$: gfg^{-1}=f\}$?

Here if we find a map $f: \Bbb P^2 \to \Bbb P^2$ birational such that $fσ=σf$ then clearly $σ \in BiRat(f)$. If possible please give me one function for 1 and then we can discuss if we can find $BiRat(f)$ or not!