We present a generalization of the $U(1)^{2}$ charged dilaton black holes family whose main feature is that both $U(1)$ fields have electric and magnetic charges, the axion field still being trivial. We show the supersymmetry of these solutions in the extreme case, in which the corresponding generalization of the Bogomolnyi bound is saturated and a naked singularity is on the verge of being visible to external observers. Then we study the action of a subset of the $SL(2,R)$ group of electric-magnetic duality rotations that generates a non-trivial axion field on those solutions. This group of transformations is an exact symmetry of the $N=4$ $d=4$ ungauged supergravity equations of motion. It has been argued recently that it could be an exact symmetry of the full effective string theory. The generalization of the Bogomolnyi bound is invariant under the full $SL(2,R)$ and the solutions explicitly rotated are shown to be supersymmetric if the originals are. We conjecture that any $SL(2,R)$ transformation will preserve supersymmetry.

22 pages