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 010^10.3 M_sun 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 Herschel PACS observations, we have an accurate, broad-band measure of the SFR for the bulk of the star-forming galaxies. We use multi-wavelength SED fitting techniques to estimate the stellar masses of all objects and the SFR of the MIPS and PACS undetected galaxies. We analyse the dependence of the SF activity, stellar mass and specific SFR on the group-centric distance, up to z~1.6, for the first time. We do not find any correlation between the mean SFR and group-centric distance at any redshift. We do not observe any strong mass segregation either, in agreement with predictions from simulations. Our results suggest that either groups have a much smaller spread in accretion times with respect to the clusters and that the relaxation time is longer than the group crossing time. ; Comment: Accepted for publication in MNRAS