"Quaternionic"vector bundles are the objects that describe topological phases of quantum systems subjected to an odd time-reversal symmetry (class AII). In this work, we prove that the Furuta-Kametani-Matsue-Minami (FKMM) invariant provides the correct fundamental characteristic class for the classification of "Quaternionic"vector bundles in dimension less than or equal to three (low dimension). The new insight is provided by the interpretation of the FKMM invariant from the viewpoint of the Bredon equivariant cohomology. This fact, along with basic results in equivariant homotopy theory, allows us to achieve the expected result.