Let $X$ be a complex manifold and let $G$ and $H$ be two finite subgroups of its automorphism group $Aut(X)$. Suppose we are given that $X/G$ and $X/H$ are bi-holomorphic complex manifolds. What can we say about $G$ and $H$?
Is it the case that $G$ and $H$ have to be isomorphic as subgroups?
The isomorphism $G\cong H$ is false already when $X=CP^1$. For every finite group $F$ of biholomorphic automorphisms of $X=CP^1$, $X/F\cong X$ (since $X/F$ is a genus zero compact connected Riemann surface).
Even if you assume that actions are free the claim is false, but to construct examples you consider an elliptic curve $X$ instead of $CP^1$. (One can find examples with $G=\{1\}$ and $H\cong Z/2\times Z/2$.)