We quantify the effect of the galaxy group environment (for group masses of 10 12.5
-1014.0 M e ) on the current star formation rate (SFR) of a pure, morphologically selected sample of disk-dominated (i.e., late-type spiral) galaxies with redshift 0.13. The sample embraces a full representation of quiescent and star-forming disks with stellar mass M * 10 9.5 M e . We focus on the effects on SFR of interactions between grouped galaxies and the putative intrahalo medium (IHM) of their host group dark matter halos, isolating these effects from those induced through galaxy-galaxy interactions, and utilizing a radiation transfer analysis to remove the inclination dependence of derived SFRs. The dependence of SFR on M * is controlled for by measuring offsets Δlog(ψ * ) of grouped galaxies about a single power-law relation in specific SFR,, exhibited by non-grouped "field" galaxies in the sample. While a small minority of the group satellites are strongly quenched, the group centrals and a large majority of satellites exhibit levels of ψ * statistically indistinguishable from their field counterparts, for all M * , albeit with a higher scatter of 0.44 dex about the field reference relation (versus 0.27 dex for the field). Modeling the distributions in Δlog(ψ * ), we find that (i) after infall into groups, disk-dominated galaxies continue to be characterized by a similar rapid cycling of gas into and out of their interstellar medium shown prior to infall, with inflows and outflows of ∼1.5-5 x SFR and ∼1-4 x SFR, respectively; and (ii) the independence of the continuity of these gas flow cycles on M * appears inconsistent with the required fueling being sourced from gas in the circumgalactic medium on scales of ∼100 kpc. Instead, our data favor ongoing fueling of satellites from the IHM of the host group halo on ∼Mpc scales, i.e., from gas not initially associated with the galaxies upon infall. 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 required substantial accretion of IHM gas by satellite disk-dominated galaxies will lead to a progressive reduction in the specific angular momentum of these systems, thereby representing an efficient secular mechanism to transform morphology from star-forming disk-dominated types to more passive spheroid-dominated types.