  In classical gravity deviations from the predictions of the Einstein theory
are often discussed within the framework of the conformal Newtonian gauge,
where scalar perturbations are described by two potentials $\phi$ and $\psi$.
In this paper we use the above gauge to explore possible cosmological
consequences of a running Newton's constant $ G (\Box) $, as suggested by the
nontrivial ultraviolet fixed point scenario arising from the quantum
field-theoretic treatment of Einstein gravity with a cosmological constant
term. Here we focus on the effects of a scale-dependent coupling on the
so-called gravitational slip functions $\eta = \psi / \phi -1 $, whose
classical general relativity value is zero. Starting from a set of manifestly
covariant but non-local effective field equations derived earlier, we compute
the leading corrections in the potentials $\phi$ and $\psi$ for a
nonrelativistic, pressureless fluid. After providing an estimate for the
quantity $\eta$, we then focus on a comparison with results obtained in a
previous paper on matter density perturbations in the synchronous gauge, which
gave an estimate for the growth index parameter $\gamma$, also in the presence
of a running $G$. Our results indicate that, in the present framework and for a
given $ G (\Box) $, the corrections tend to be significantly larger in
magnitude for the perturbation growth exponents than for the conformal
Newtonian gauge slip function.
