  Previous work in the literature had built a formalism for spatially averaged
equations for the scale factor, giving rise to an averaged Raychaudhuri
equation and averaged Hamiltonian constraint, which involve a backreaction
source term. The present paper extends these equations to include models with
variable Newton parameter and variable cosmological term, motivated by the
nonperturbative renormalization program for quantum gravity based upon the
Einstein-Hilbert action. We focus on the Brans-Dicke form of the
renormalization-group improved action functional. The coupling between
backreaction and spatially averaged three-dimensional scalar curvature is found
to survive, and a variable-G cosmic quintet is found to emerge. Interestingly,
under suitable assumptions, an approximate solution can be found where the
early universe tends to a FLRW model, while keeping track of the original
inhomogeneities through three effective fluids. The resulting qualitative
picture is that of a universe consisting of baryons only, while inhomogeneities
average out to give rise to the full dark-side phenomenology.
