# Holonomy groups of quotient Riemannian manifolds?

Let $(X,g)$ be a Riemannian manifold with holonomy group $Hol(X,g)$. Suppose that a finite group $G$ acts on $X$ freely and the metric $g$ is invariant under $G$. What can one say about the the holonomy group of the Riemannian manifold $(X/G,\tilde{g})$, where $\tilde{g}$ is the metric induced by $g$?

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...and, furthermore, the holonomy group of the quotient contains the original one as a finite index subgroup (index is at most $|G|$). –  Misha Apr 27 '13 at 10:05