Can Calabi-Yau manifolds have nonabelian discrete symmetry groups? A particle physicist asked me the above question.  Let me try to make it more precise.  Suppose $M$ is a 3-dimensional Calabi-Yau manifold: that is, a compact Kähler manifold of complex dimension 3 whose holonomy group is contained in $\mathrm{SU}(3)$.  Let $\mathrm{Aut}(M)$ be its group of holomomorphic metric-preserving diffeomorphisms.  What can this group be like?  In particular: 
1) which nonabelian discrete groups can $\mathrm{Aut}(M)$ contain?
or if that's unmanageable:
2) which nonabelian discrete groups can appear as the group of connected components of $\mathrm{Aut}(M)$?
I believe he is particularly curious as to whether we can get $\mathrm{PSL}(2,7)$ as the answer to either of these questions.
 A: If you work one dimension down, at the level of K3 surfaces, there's a very pretty classification of finite group actions preserving the holomorphic form due to Mukai.  In that classification, the simple group of order 168 is extremal.  Oguiso and Zhang have a nice article on the properties of K3 surfaces which admit such an action.  To be specific, you can get a K3 surface with this group action (the "Klein-Mukai surface") by taking a fourfold cover of the plane branched over Klein's quartic: $x^3 y + y^3 z + z^3 x + w^4=0$.  It would be fun to look at Calabi-Yau threefolds with a geometric relationship to the Klein-Mukai surface.
A: The main theorem of Fine and Panov's 'The diversity of symplectic Calabi-Yau six-manifolds'   (http://arxiv.org/abs/1108.5944) implies that, in particular, every finitely presented group arises as the fundamental group of a Calabi--Yau. As in the Kaehler case, what you want then follows by passing to the universal cover.
Edit: As abx points out, Fine and Panov construct symplectic Calabi--Yaus, which are not necessarily complex Calabi--Yaus as required by the question.  But I'll leave this answer up, as I suspect that this kind of construction may be what's needed.  Both Fine and Panov are sometimes active on MO, so perhaps one of them will answer.
A: Assuming you actually meant "Kähler" and not "Calabi-Yau": 
In the book Fundamental Groups of Compact Kähler Manifolds by Amorós et al., on page 6 (example 1.11) it is asserted that every finite group can occur as the fundamental group of a compact Kähler manifold, and the result is attributed to Serre (J.P. Serre, Sur la topologie des variétés algébriques en caractéristique p).
Hence, by taking covers, it can also occur as a subgroup of holomorphic isometries of a compact Kähler manifold.
