In the local Universe, galaxy properties show a strong dependence on environment. In cluster cores, early-type galaxies dominate, whereas star-forming galaxies are more and more common in the outskirts. At higher redshifts and in somewhat less dense environments (e.g. galaxy groups), the situation is less clear. One open issue is that of whether and how the star formation rate (SFR) of galaxies in groups depends on the distance from the centre of mass. To shed light on this topic, we have built a sample of X-ray selected galaxy groups at 0 < z < 1.6 in various blank fields [Extended Chandra Deep Field South (ECDFS), Cosmological Evolution Survey (COSMOS), Great Observatories Origin Deep Survey (GOODS)]. We use a sample of spectroscopically confirmed group members with stellar mass M* > 1010.3M* in order to have a high spectroscopic completeness. As we use only spectroscopic redshifts, our results are not affected by uncertainties due to projection effects. We use several SFR indicators to link the star formation (SF) activity to the galaxy environment. Taking advantage of the extremely deep mid-infrared Spitzer MIPS and far-infrared Herschel1 PACS observations, we have an accurate, broad-band measure of the SFR for the bulk of the star-forming galaxies. We use multi-wavelength Spectral Energy Distribution (SED) fitting techniques to estimate the stellar. © 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.