Context. Groups of galaxies are a common environment, bridging the gap between starforming field galaxies and quiescent cluster galaxies. Within groups secular processes could be at play, contributing to the observed strong decrease of star formation with cosmic time in the global galaxy population. Aims. We took advantage of the wealth of information provided by the first ∼10 000 galaxies of the zCOSMOS-bright survey and its group catalogue to study in detail the complex interplay between group environment and galaxy properties. Methods. The classical indicator F blue , i.e., the 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, divided as to define groups/field/isolated galaxy subsamples. Results. Using rest-frame evolving B-band volume-limited samples, the groups galaxy population exhibits significant blueing as redshift increases, but maintains a systematic difference (a lower F blue ) with respect to the global galaxy population, and an even larger difference with respect to 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 carefully focusing the analysis on narrow mass bins such that mass segregation becomes negligible we find that only for the lowest mass bin explored, i.e., log(M * /M ) ≤ 10.6, does a significant residual difference in color remain as a function of environment, while this difference becomes negligible toward higher masses. Conclusions. Our results indicate that red galaxies of mass log(M * /M ) ≥ 10.8 are already in place at z ∼ 1 and do not exhibit any strong environmental dependence, possibly originating from so-called nature or 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. Our analysis implies that these mechanisms begin to significantly influence galaxy evolution after z ∼ 1, a redshift corresponding to the emergence of structures in which these mechanisms take place.