We classify T²-GKM fibrations in which both fiber and base are the GKM graph of S⁴, with standard weights in the base. For each case in which the total space is orientable, we construct, by explicit clutching, a realization as a T²-equivariant linear S⁴-bundle over S⁴. We determine which of the total spaces of these examples are non-equivariantly homotopy equivalent, homeomorphic or diffeomorphic, thereby finding many examples of a) pairs of homotopy equivalent, non-homeomorphic GKM manifolds with different first Pontryagin class, and b) pairs of GKM actions on the same smooth manifold whose GKM graphs do not agree as unlabeled graphs.
Goertsches et al. (Wed,) studied this question.