Supermassive black holes (BHs) residing in the brightest cluster galaxies are over-massive relative to the stellar bulge mass or central stellar velocity dispersion of their host galaxies. As BHs residing at the bottom of the galaxy clusters potential well may undergo physical processes that are driven by the large-scale characteristics of the galaxy clusters, it is possible that the growth of these BHs is (indirectly) governed by the properties of their host clusters. In this work, we explore the connection between the mass of BHs residing in the brightest group/cluster galaxies (BGGs/BCGs) and the virial temperature, and hence total gravitating mass, of galaxy groups/clusters. To this end, we investigate a sample of 17 BGGs/BCGs with dynamical BH mass measurements and utilize XMM-Newton X-ray observations to measure the virial temperatures and infer the $M_{rm 500}$ mass of the galaxy groups/clusters. We find that the $M_{rm BH} - kT$ relation is significantly tighter and exhibits smaller scatter than the $M_{rm BH} - M_{rm bulge}$ relations. The best-fitting power-law relations are $ log_{10} (M_{rm BH}/10^{9} rm{M_{odot}}) = 0.20 + 1.74 log_{10} (kT/1 rm{keV}) $ and $ log_{10} (M_{rm BH}/10^{9} rm{M_{odot}}) = -0.80 + 1.72 log_{10} (M_{rm bulge}/10^{11} M_{odot})$. Thus, the BH mass of BGGs/BCGs may be set by physical processes that are governed by the properties of the host galaxy group/cluster. These results are confronted with the Horizon-AGN simulation, which reproduces the observed relations well, albeit the simulated relations exhibit notably smaller scatter.
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