Let $X$ be a projective complex manifold and $G$ a finite group. Assume that $G$ acts on $X$ holomorphically and freely. Is it true that any birational map $\phi \in Bir(X/G)$ lifts to some birational map $\tilde{\phi}\in Bir(X)$? If this is not true, what kind of condition should one impose on the manifold or the cation?

Let $X$ be a projective complex manifold and $G$ a finite group. Assume that $G$ acts on $X$ holomorphically and freely. Is it true that any birational map $\phi \in Bir(X/G)$ lifts to some birational map $\tilde{\phi}\in Bir(X)$? If this is not true, what kind of condition should one impose on the manifold or the action?