Does the metric of a homogeneous compact metric space lift to a metric on its group of isometries?

Question

Let $X$ be a compact metric space, $Iso(X)$ be its group of isometries, and assume further that the action of $Iso(X)$ on $X$ is transitive. If we take the stabilizer $G_x$ of any element $x \in X$, we have that $X$ can be identified with $Iso(X)/G_x$. Now, does the metric $d = d_X$ on $X$ lift to a metric on $Iso(X)$?