  In this note, we consider unramified cohomology with $\mathbb{Z}/2$
coefficients for some (degree two) quotient varieties and describe a method
that allows one to prove the non-vanishing of these groups under certain
conditions. We apply this method to prove a non-vanishing statement in the case
of Kummer varieties. Combining this with work of Colliot-Th\'el\`ene and
Voisin, we obtain a new type of three-dimensional counterexample to the
integral Hodge conjecture.
