  We study the evolution of the total star formation (SF) activity, total
stellar mass and halo occupation distribution in massive halos by using one of
the largest X-ray selected sample of galaxy groups with secure spectroscopic
identification in the major blank field surveys (ECDFS, CDFN, COSMOS, AEGIS).
We provide an accurate measurement of SFR for the bulk of the star-forming
galaxies using very deep mid-infrared Spitzer MIPS and far-infrared Herschel
PACS observations. For undetected IR sources, we provide a well-calibrated SFR
from SED fitting. We observe a clear evolution in the level of SF activity in
galaxy groups. The total SF activity in the high redshift groups (0.5<z<1.1) is
higher with respect to the low redshift (0.15<z<0.5) sample at any mass by
0.8+/-0.12 dex. A milder difference (0.35+/-0.1 dex) is observed between the
low redshift bin and the groups at z~0. We show that the level of SF activity
is declining more rapidly in the more massive halos than in the more common
lower mass halos. We do not observe any evolution in the halo occupation
distribution and total stellar mass- halo mass relations in groups. The picture
emerging from our findings suggests that the galaxy population in the most
massive systems is evolving faster than galaxies in lower mass halos,
consistently with a "halo downsizing" scenario.
