This article summarizes the state of the art of $ν_μ$ and $\barν_μ$ CC0$π$ cross-section measurements on carbon and argon and discusses the relevant nuclear models, parametrizations and uncertainties in GENIE v3. The CC0$π$ event topology is common in experiments at a few-GeV energy range. Although its main contribution comes from quasi-elastic interactions, this topology is still not well understood. The GENIE global analysis framework is exploited to analyze CC0$π$ datasets from MiniBooNE, T2K and MINERvA. A partial tune for each experiment is performed, providing a common base for the discussion of tensions between datasets. The results offer an improved description of nuclear CC0$π$ datasets as well as data-driven uncertainties for each experiment. This work is a step towards a GENIE global tune that improves our understanding of neutrino interactions on nuclei. It follows from earlier GENIE work on the analysis of neutrino scattering datasets on hydrogen and deuterium.