We took advantage of the wealth of information provided by the first ~10000 galaxies of the zCOSMOS-bright survey and its group catalogue to study the complex interplay between group environment and galaxy properties. The classical indicator F_blue (fraction of blue galaxies) proved to be a simple but powerful diagnostic tool. We studied its variation for different luminosity and mass selected galaxy samples. Using rest-frame B-band selected samples, the groups galaxy population exhibits significant blueing as redshift increases, but maintains a lower F_blue with respect both to the global and the isolated galaxy population. However moving to mass selected samples it becomes apparent that such differences are largely due to the biased view imposed by the B-band luminosity selection, being driven by the population of lower mass, bright blue galaxies for which we miss the redder, equally low mass, counterparts. By focusing the analysis on narrow mass bins such that mass segregation becomes negligible we find that only for the lowest mass bin explored (logMass <= 10.6) does a significant residual difference in color remain as a function of environment, while this difference becomes negligible toward higher masses. Our results indicate that red galaxies of logMass >= 10.8 are already in place at z ~ 1 and do not exhibit any strong environmental dependence, possibly originating from so-called 'nature'/internal mechanisms. In contrast, for lower galaxy masses and redshifts lower than z ~ 1, we observe the emergence in groups of a population of 'nurture' red galaxies: slightly deviating from the trend of the downsizing scenario followed by the global galaxy population, and more so with cosmic time. These galaxies exhibit signatures of group-related secular physical mechanisms directly influencing galaxy evolution. Comment: Submitted to A&A, revised version after referee comments