  The co-evolution between supermassive black holes and their environment is
most directly traced by the hot atmospheres of dark matter halos. Cooling of
the hot atmosphere supplies the central regions with fresh gas, igniting active
galactic nuclei (AGN) with long duty cycles. Outflows from the central engine
tightly couple with the surrounding gaseous medium and provide the dominant
heating source preventing runaway cooling. Every major modern hydrodynamical
simulation suite now includes a prescription for AGN feedback to reproduce
realistic populations of galaxies. However, the mechanisms governing the
feeding/feedback cycle between the central black holes and their surrounding
galaxies and halos are still poorly understood. Galaxy groups are uniquely
suited to constrain the mechanisms governing the cooling-heating balance, as
the energy supplied by the central AGN can exceed the gravitational binding
energy of halo gas particles. Here we provide a brief overview of our knowledge
of the impact of AGN on the hot atmospheres of galaxy groups, with a specific
focus on the thermodynamic profiles of groups. We then present our on-going
efforts to improve on the implementation of AGN feedback in galaxy evolution
models by providing precise benchmarks on the properties of galaxy groups. We
introduce the \XMM~ Group AGN Project (X-GAP), a large program on \XMM~
targeting a sample of 49 galaxy groups out to $R_{500c}$.
