Is it true that for every finite $G\leq Aut(\mathbb{P}^1_{\mathbb{C}})=PGL_2(\mathbb{C})$ the morphism $\mathbb{P}^1_{\mathbb{C}}\rightarrow \mathbb{P}^1_{\mathbb{C}}/G$ descends as a morphism (not nec. with the group actions) to $\mathbb{Q}$?
My intuition is going haywire here. For a while I think it's true, and then I think it's not. Do you have a decisive answer?