The method representative_action() for Sage permutation groups erroneously returns the identity element when it actually should either fail or return something which makes clear if no representative exists. The corresponding method for libgap groups behaves as expected:
sage: g = PermutationGroup([[(1, 2)], [(3, 4)]])
sage: g.representative_action(1, 3)
()
sage: libgap(g).RepresentativeAction(1, 3)
fail