The construction of Kummer K3 surfaces from abelian surfaces can be generalized to yield higher dimensional varieties known as hyperk"ahler varieties of Kummer type. Hassett and Tschinkel showed that a portion of the middle cohomology of generalized Kummer 4-folds may be understood as fixed loci of symplectic involutions corresponding to the three-torsion points of the abelian surface. In recent work with Sarah Frei, we have extended this result, allowing us to characterize the Galois action on the cohomology when working over non-closed fields.