One of the most fundamental correlations between the properties of galaxies in the local Universe is the so-called morphology-density relation (Dressler 1980). A plethora of studies utilizing multi-wavelength tracers of activity have shown that late type star forming galaxies favour low density regions in the local Universe (e.g. G´omez et al. 2003). In particular, the cores of massive galaxy clusters are galaxy graveyards full of massive spheroids that are dominated by old stellar populations. A variety of physical processes might be effective in suppressing star formation and affecting the morphology of cluster and group galaxies. Broadly speaking, these can be grouped in two big families: (i) interactions with other cluster members and/or with the cluster gravitational potential and (ii) interactions with the hot gas that permeates massive galaxy systems. Galaxy groups are the most common galaxy environment in our Universe, bridging the gap between the low density field and the crowded galaxy clusters. Indeed, as many as 50%-70% of galaxies reside in galaxy groups in the nearby Universe (Huchra & Geller 1982; Eke et al. 2004), while only a few percent are contained in the denser cluster cores. In addition, in the current bottom-up paradigm of structure formation, galaxy groups are the building blocks of more massive systems: they merge to form clusters. As structures grow, galaxies join more and more massive systems, spending most of their life in galaxy groups before entering the cluster environment. Thus, it is plausible to ask if group-related processes may drive the observed relations between galaxy properties and their environment. To shed light on this topic we have built the largest X-ray selected samples of galaxy groups with secure spectroscopic identification on the major blank field surveys. For this purpose, we combine deep X-ray Chandra and XMM data of the four major blank fields (All-wavelength Extended Groth Strip International Survey (AEGIS), the COSMOS field, the Extended Chandra Deep Field South (ECDFS), and the Chandra Deep Field North (CDFN) ). The group catalog in each field is created by associating any X-ray extended emission to a galaxy overdensity in the 3D space. This is feasible given the extremely rich spectroscopic coverage of these fields. Our identification method and the dynamical analysis used to identify the galaxy group members and to estimate the group velocity dispersion is extensively tested on the AEGIS field and with mock catalogs extracted from the Millennium Simulation (Springel et al. 2005). The effect of dynamical complexity, substructure, shape of X-ray emission, different radial and redshift cuts have been explored on the LX −sigma relation. We also discover a high redshift group at z~1.54 in the AEGIS field. This detection illustrates that mega-second Chandra exposures are required for detecting such objects in the volume of deep fields. We provide an accurate measure of the Star Formation Rate (SFR) of galaxies by using the deepest available Herschel PACS and Spitzer MIPS data available for the considered fields. We also provide a well-calibrated estimate of the SFR derived by using the SED fitting technique for undetected sources in mid- and far-infrared observations. Using this unique sample, we conduct a comprehensive analysis of the dependence of the total SFR , total stellar masses and halo occupation distribution (HOD) of massive galaxies (M*>10^10 M_sun) on the halo mass of the groups with rigorous consideration of uncertainties. We observe a clear evolution in the level of star formation (SF) activity in galaxy groups. Indeed, the total star formation activity in high redshift (0.5 10^10.4−10.6 M_sun). Above this limit, the galaxy SFR has a very weak dependence on the stellar mass. This flattening, to different extent, is present in all environments. At low redshift, group galaxies tend to deviate more from the mean MS towards the region of quiescence with respect to isolated and filament-like galaxies. This environment dependent location of low redshift group galaxies with respect to the mean MS causes the increase of the dispersion of the distribution of galaxies around the MS as a function of the stellar mass. At high redshift we do not find significant evidence for a differential location of galaxies with respect to the MS as a function of the environment. Indeed, in this case we do not observe a significant increase of the dispersion of the distribution of galaxies around the MS as a function of the stellar mass. We do not find evidence for a differential distribution in the morphological type of MS galaxies in different environments. Instead, we observe a much stronger dependence of the mean S´ersic index on the stellar mass. These results suggest that star formation quenching in group galaxies is not due to galaxy structural transformations. It also suggests that while morphology of MS galaxies is more stellar mass dependent, star formation quenching is mostly environment dependent. We conclude that the membership to a massive halo is a key ingredient in the galaxy evolution and that this acts in terms of star formation quenching in group sized halos.