Finite quotients of abelian varieties with a Calabi-Yau resolution

Let $A$ be an abelian variety, and $G \subset Aut(A)$ a finite group acting freely in codimension two. We discuss whether the singular quotient $A/G$ admits a resolution that is a Calabi-Yau manifold. While Oguiso constructed two examples in dimension $3$, we show that there are none in dimension $4$. We also classify up to isogeny the possible abelian varieties $A$ in arbitrary dimension.
Comment: [v1]: 54 pages, including the programs in the 6-page-long Appendix. [v2]: final version, updated after referee report. Theorem 1.5 is slightly more general now