  General Relativity extensions based on Renormalization Group effects are
motivated by a known physical principle and constitute a class of extended
gravity theories that have some unexplored unique aspects. In this work we
develop in detail the Newtonian and post Newtonian limits of a realisation
called Renormalization Group extended General Relativity (RGGR). Special
attention is taken to the external potential effect, which constitutes a type
of screening mechanism typical of RGGR. In the Solar System, RGGR depends on a
single dimensionless parameter $\bar \nu_\odot$, and this parameter is such
that for $\bar \nu_\odot = 0$ one fully recovers GR in the Solar System.
Previously this parameter was constrained to be $|\bar \nu_\odot| \lesssim
10^{-21}$, without considering the external potential effect. Here we show that
under a certain approximation RGGR can be cast in a form compatible with the
Parametrised Post-Newtonian (PPN) formalism, and we use both the PPN formalism
and the Laplace-Runge-Lenz technique to put new bounds on $\bar \nu_\odot$,
either considering or not the external potential effect. With the external
potential effect the new bound reads $|\bar \nu_\odot| \lesssim 10^{-16}$. We
discuss the possible consequences of this bound to the dark matter abundance in
galaxies.
