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