Black holes and equivariant charge vectors in N=2,d=4 supergravity

We extend previous investigations on the construction of extremal supersymmetric and non-supersymmetric solutions in the H-FGK formalism to "unconventional" solutions with anharmonic terms. We show how the use of "fake" charge vectors equivariant under duality transformations simplifies and clarifies the task of identification of the attractors of the theory.