A brief review of the essentials of Asymptotic Safety and the Renormalization Group (RG) improvement of the Schwarzschild Black Hole that removes the $ r = 0$ singularity is presented. It is followed with a RG-improvement of the Kantowski-Sachs metric associated with a Schwarzschild black hole interior and such that there is $no$ singularity at $ t = 0$ due to the running Newtonian coupling $ G ( t )$ (vanishing at $ t = 0$). Two temporal horizons at $ t _- \simeq t_P$ and $ t_+ \simeq t_H$ are found. For times below the Planck scale $ t < t_P$, and above the Hubble time $ t > t_H$, the components of the Kantowski-Sachs metric exhibit a key sign $change$, so the roles of the spatial $z$ and temporal $t$ coordinates are $exchanged$, and one recovers a $repulsive$ inflationary de Sitter-like core around $ z = 0$, and a Schwarzschild-like metric in the exterior region $ z > R_H = 2 G_o M $. The inclusion of a running cosmological constant $ \Lambda (t) $ follows. We proceed with the study of a dilaton-gravity (scalar-tensor theory) system within the context of Weyl's geometry that permits to single out the expression for the $classical$ potential $ V (\phi ) = \kappa\phi^4$, instead of being introduced by hand, and find a family of metric solutions which are conformally equivalent to the (Anti) de Sitter metric. To conclude, an ansatz for the truncated effective average action of ordinary dilaton-gravity in Riemannian geometry is introduced, and a RG-improved Cosmology based on the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is explored where instead of recurring to the cutoff identification $ k = k ( t ) = \xi H ( t ) $, based on the Hubble function $ H (t)$, with $ \xi $ a positive constant, one has now $ k = k ( t ) = \xi \phi ( t ) $, when $ \phi $ is a positive-definite dilaton scalar field which is monotonically decreasing with time.