By a Calabi-Yau variety I mean a smooth projective variety over the complex numbers with numerically trivial canonical divisor.

For each postive integer $n$ does there exist a finite group $G$ (possibly depending on $n$) which is not the fundamental group of a Calabi-Yau variety of dimension $n$?

For $n=1,2$ this follows easily from the classification of Calabi-Yau varieties of these dimensions, the only non-trivial finite fundamental group being $\mathbb{Z}/2\mathbb{Z}$ (for Enriques surfaces). For $n=3$ some finite non-abelian groups are known to occur as fundamental groups but I do not know of any non-existence results.

If there are only finitely many families of Calabi-Yau varieties of a given dimension then the question would clearly have a positive answer. However, this is far from being known so I am interested in other possible approaches.