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The iron K$\alpha$ line commonly observed in the X-ray spectrum of bothstellar-mass and supermassive black hole candidates originates from X-rayfluorescence of the inner accretion disk. Accordingly, it can be used to mapthe spacetime geometry around these objects. In this paper, we extend previouswork using the iron K$\alpha$ line to test the Kerr black hole hypothesis. Weadopt the Cardoso-Pani-Rico parametrization and we test the possibility ofconstraining possible deviations from the Kerr solution that can be obtainedfrom observations across the range of black hole spins and inclination angles.We confirm previous claims that the iron K$\alpha$ line is potentially a quitepowerful probe for testing the Kerr metric given sufficiently high quality dataand with systematics under control, especially in the case of fast-rotatingblack holes and high inclination angles since both conditions serve to maximizerelativistic effects. We find that some geometric perturbations from Kerrgeometry manifest more strongly in the iron line profile than others. While theperturbation parameter $\epsilon^t_3$ can be well constrained by the iron lineprofile, an orthogonal data set is necessary to constrain departures from Kerrgeometry in $\epsilon^r_3$.

Testing the Kerr Nature of Black Hole Candidates using Iron Line Spectra in the CPR Framework