  As their smooth analogue the irreducible symplectic varieties appear as
elementary bricks in the generalizations of the Bogomolov decomposition theorem
(arXiv:math/0402243, arXiv:2012.00441). Let $S$ be a K3 surface; generalizing
the Fujiki construction, we investigate the irreducible symplectic varieties
with simply connected smooth locus that can be obtained as terminalizations of
quotients of the product $S^{n}$. In dimension 4, we compute the singularities
for 29 orbifolds examples which appear to be independent under deformation. We
also provide 4 additional orbifolds examples in dimension 6.
