Abridged - We quantify the effect of the galaxy group environment (for 12.5 9.5. Within this population, we find that, while a small minority of group satellites are strongly quenched, the group centrals, and the large majority of satellites exhibit levels of SFR indistinguishable from ungrouped "field" galaxies of the same M*, albeit with a higher scatter, and for all M*. Modelling these results, we deduce that disk-dominated satellites continue to be characterized by a rapid cycling of gas into and out of their ISM at rates similar to those operating prior to infall, with the on-going fuelling likely sourced from the group intrahalo medium (IHM) on Mpc scales, rather than from the circum-galactic medium on 100kpc scales. Consequently, the color-density relation of the galaxy population as a whole would appear to be primarily due to a change in the mix of disk- and spheroid-dominated morphologies in the denser group environment compared to the field, rather than to a reduced propensity of the IHM in higher mass structures to cool and accrete onto galaxies. We also suggest that the inferred substantial accretion of IHM gas by satellite disk-dominated galaxies will lead to a progressive reduction in their specific angular momentum, thereby representing an efficient secular mechanism to transform morphology from star-forming disk-dominated types to more passive spheroid-dominated types. ; Comment: Accepted for publication in AJ; 40 pages, 27 figures (8 full page), 6 tables, 5 appendices (10 pages), data in figures available in machine readable format from journal (or author on demand)