TITLE: Quotients of Abelian varieties by finite groups
QUESTION [9 upvotes]: Let $A$ be an Abelian variety (over an algebraically closed field).  The group $\mathbb{Z}/2\mathbb{Z}$ acts on $A$ and the quotient is called the Kummer variety of $A$.  These are well studied and, as I understand it, their geometry is understood and a classical subject (especially for surfaces).
What happens when you take the quotient of an Abelian variety by other finite groups?  Is the geometry of the quotient understood in any sense (e.g., Kodaira dimension)?  Are the quotients somehow classified?  Is anything else known about them?

REPLY [2 votes]: If you are still interested at the moment there is a complete classification in 3 and 4 dimension when the group acts freely on the abelian varieties (they are called hyperelliptic varieties). Some authors are Catanese and Demleitner: if you search their papers you can find others (and better) references.
Maybe you are also interested in the so called generalized kummer.