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X-ray reflection spectroscopy is a promising technique for testing generalrelativity in the strong field regime, as it can be used to test the Kerr blackhole hypothesis. In this context, the parametrically deformed black holemetrics proposed by Konoplya, Rezzolla \& Zhidenko (Phys. Rev. D93, 064015,2016) form an important class of non-Kerr black holes. We implement this classof black hole metrics in \textsc{relxill\_nk}, which is a framework we havedeveloped for testing for non-Kerr black holes using X-ray reflectionspectroscopy. We perform a qualitative analysis of the effect of the leadingorder strong-field deformation parameters on typical observables like theinnermost stable circular orbits and the reflection spectra. We also presentthe first X-ray constraints on some of the deformation parameters of thismetric, using \textit{Suzaku} data from the supermassive black hole in Ark~564,and compare them with those obtained (or expected) from other observationaltechniques like gravitational waves and black hole imaging.

Testing General Relativity with X-ray reflection spectroscopy: The Konoplya-Rezzolla-Zhidenko parametrization