Recently, relations connecting the SMBH mass of central galaxies and global properties of the hosting cluster, such as temperature and mass, were observed. We investigate the correlation between SMBH mass and cluster mass and temperature, their establishment and evolution. We compare their scatter to that of the classical $M_{rm BH}-M_{rm BCG}$ relation. We study how gas accretion and BH-BH mergers contribute to SMBH growth across cosmic time. We employed 135 groups and clusters with a mass range $1.4times 10^{13}M_{odot}-2.5times 10^{15} M_{odot}$ extracted from a set of 29 zoom-in cosmological hydro-dynamical simulations where the baryonic physics is treated with various sub-grid models, including feedback by AGN. In our simulations we find that $M_{rm BH}$ correlates well with $M_{500}$ and $T_{500}$, with the scatter around these relations compatible within $2sigma$ with the scatter around $M_{rm BH}-M_{rm BCG}$ at $z=0$. The $M_{rm BH}-M_{500}$ relation evolves with time, becoming shallower at lower redshift as a direct consequence of hierarchical structure formation. On average, in our simulations the contribution of gas accretion to the total SMBH mass dominates for the majority of the cosmic time ($z>0.4$), while in the last 2 Gyr the BH-BH mergers become a larger contributor. During this last process, substructures hosting SMBHs are disrupted in the merger process with the BCG and the unbound stars enrich the diffuse stellar component rather than increase BCG mass. From the results obtained in our simulations with simple sub-grid models we conclude that the scatter around the $M_{rm BH}-T_{500}$ relation is comparable to the scatter around the $M_{rm BH}-M_{rm BCG}$ relation and that, given the observational difficulties related to the estimation of the BCG mass, clusters temperature and mass can be a useful proxy for the SMBHs mass, especially at high redshift.
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