The global minima of the cluster anions with the generic chemical formula (XAl12)(2-), where X = Be, Mg, Ca, Sr, Ba, and Zn, are determined by an extensive search of their potential energy surfaces using the Gradient Embedded Genetic Algorithm (GEGA). All the characterized global minima have an icosahedral-like structure, resembling that of the Al-13(-) cluster. These cages comprise closed-shell electronic configurations with 40 electrons, therefore, in accordance to the jellium model, they are predicted to be highly stable and amenable to experimental detection. The two preferred sites for the dopant species, at the center and at surface of the icosahedral cage, are stabilized depending on the atomic radius of X Thus, while the small dopants (X = Be, Zn) sit preferably at the center of the cage, the preferred site for X = Mg, Ca, Sr, and Ba is at the surface. Since these dianions are not stable towards electron detachment, one Li cation is added in order to yield stable systems. Our computations show that in the global minimum form of Li(XAl12), the lithium cation, ionically bonded to the Al atoms, does not change the structure of the (XAl12)2 core.