Coarse median spaces and groups were introduced by Bowditch in 2013. Recently, Fioravanti studied automorphisms for which the coarse median of every triple is mapped to some point close to the coarse median of the images with great success. In this talk, we will discuss how this property studied by Fioravanti relates to some other well-known properties of homomorphisms of groups. We will mostly focus on hyperbolic groups, but we might make a detour to some other subclasses of right-angled Artin groups.