  Models of $f(R)$ gravity that introduce corrections to the Newtonian
potential in the weak field limit are tested at the scale of galaxy clusters.
These models can explain the dynamics of spiral and elliptical galaxies without
resorting to dark matter. We compute the pressure profiles of 579 galaxy
clusters assuming that the gas is in hydrostatic equilibrium within the
potential well of the modified gravitational field. The predicted profiles are
compared with the average profile obtained by stacking the data of our cluster
sample in the Planck foreground clean map SMICA. We find that the resulting
profiles of these systems fit the data without requiring a dominant dark matter
component, with model parameters similar to those required to explain the
dynamics of galaxies. Our results do not rule out that clusters are dynamically
dominated by Dark Matter but support the idea that Extended Theories of Gravity
could provide an explanation to the dynamics of self-gravitating systems and to
the present period of accelerated expansion, alternative to the concordance
cosmological model.
