Holonomy group of quotient manifold 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 $Hol(M,g_M)\cong Hol(M/G,g_{M/G})$?

Perhaps you can get an idea of the difficulty of your question by considering quotients of a flat torus. Have a look of the following papers:
http://terpconnect.umd.edu/~wmg/Bieberbach.pdf 
http://mat.ug.edu.pl/~aszczepa/problemsfv.pdf
cheers,
h.
