On the FOD/FOM parameter of rational maps

Let $\chi$ be a (right) action of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{L})$ on the space $\mathbb{L}(z)$ of rational maps defined over an algebraically closed field $\mathbb{L}$. If $R\in \mathbb{L}(z)$ and ${\mathcal{M}}_R^{\chi }$ is its $\chi$-field of moduli, then the parameter ${\mathrm{F}\mathrm{O}\mathrm{D}\mathrm{/}\mathrm{F}\mathrm{O}\mathrm{M}}_{\chi }(R)$ is the smallest integer $n\ge 1$ such that there is a $\chi$-field of definition of $R$ being a degree $n$ extension of ${\mathcal{M}}_R^{\chi }$. When $\mathbb{L}$ has characteristic zero and $\chi ={\chi }_{\mathrm{\infty }}$ is the conjugation action, then it is known that ${\mathrm{F}\mathrm{O}\mathrm{D}\mathrm{/}\mathrm{F}\mathrm{O}\mathrm{M}}_{{\chi }_{\mathrm{\infty }}}(R)\le 2$. In this paper, we study the above parameter for general actions and any characteristic.