Let $(M,g_M)$ be a compact Riemannian manifold with holonomy group $Hol(M,g_M)$. Suppose that a finite group $G$ acts on $M$ freely and preserves the metric $g$.
What can one say about the holonomy group $Hol(M/G,g_{M/G})$ of $M/G$ equipped with the induced metric $g_{M/G}$?
Are there any good conditions to guarantee $Hol(M,g_M)\cong Hol(M/G,g_{M/G})$?