It is widely known that the Kepler problem admits SU(2,2) as a dynamical group. The author aims to show that SU(2,2) is also a dynamical group for the MIC-Kepler problem, a generalization of the Kepler problem. It is already known that the symmetry groups for the MIC-Kepler problem are SO(4), SO0(1,3), and E(3), according to whether the energy is negative, positive, or zero. It is shown that the double cover of the respective symmetry groups, SU(2)*SU(2), SL(2,C), and SU(2)*R3, a semi-direct product, are realized as subgroups of SU(2,2). Isoenergetic orbit spaces are also studied, which are defined to be quotient manifolds of the respective energy manifolds by the respective Hamiltonian flows. Each of the isoenergetic orbit spaces is shown to be realized as a (co-)adjoint orbit of the symmetry group. In addition, use of the isoenergetic orbit spaces is discussed. In fact, a certain class of perturbed MIC-Kepler problems are shown to induce dynamical systems on the isoenergetic orbit space. If the energy is negative, the generic isoenergetic orbit space is diffeomorphic with S2*S2, so that the Euler number of S2*S2 provides the number of singular points for the reduced perturbed system, and in turn that of closed orbits for the perturbed MIC-Kepler problem.