  Here we apply the full Will-Nordtvedt version of the Parameterized
Post-Newtonian (PPN) formalism to a class of General Relativity extensions that
are based on nontrivial renormalization group (RG) effects at large scales. We
focus on a class of models in which the gravitational coupling constant $G$ is
correlated with the Newtonian potential. A previous PPN analysis considered a
specific realization of the RG effects, and only within the
Eddington-Robertson-Schiff version of the PPN formalism, which is a less
complete and robust PPN formulation. Here we find stronger, more precise
bounds, and with less assumptions. We also consider the External Potential
Effect (EPE), which is an effect that is intrinsic to this framework and
depends on the system environment (it has some qualitative similarities to the
screening mechanisms of modified gravity theories). We find a single particular
RG realization that is not affected by the EPE. Some physical systems have been
pointed out as candidates for measuring the possible RG effects in gravity at
large scales, for any of them the Solar System bounds need to be considered.
