Do elements of the fundamental group give rise to isometries

Do elements of the fundamental group give rise to isometries

Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its
universal cover. Suppose that there exists a Kahler-Einstein metric on
$\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
Question. Assume $X$ is compact. Is $\pi_1(X)$ actually a subgroup of the
group of isometries of $\tilde X$ with respect to the Kahler-Einstein
metric?
I suspect it is, but I don't know enough about analytic geometry to be
completely sure.