We use Chandra observations of 13 nearby groups of galaxies to investigate the hot gas content of their member galaxies. We find that a large fraction of near-IR-bright, early-type galaxies in groups have extended X-ray emission, indicating that they retain significant hot gas halos even in these dense environments. In particular, we detect hot gas halos in ～80% of L_K > L_* galaxies. We do not find a significant difference in the L_K-L_X relation for detected group and cluster early-type galaxies. However, we detect X-ray emission from a significantly higher fraction of galaxies brighter than L_* in groups compared to clusters, indicating that a larger fraction of galaxies in clusters experience significant stripping of their hot gas. In addition, group and cluster galaxies appear to be X-ray-faint compared to field galaxies, although a Chandra-based field sample is needed to confirm this result. The near-IR-bright late-type galaxies in clusters and groups appear to follow the L_K-L_X relation for early-type galaxies, while near-IR-fainter late-type galaxies are significantly more X-ray luminous than this relation likely due to star formation. Finally, we find individual examples of ongoing gas stripping of group galaxies. One galaxy shows a 40-50 kpc X-ray tail, and two merging galaxy systems show tidal bridges/tails of X-ray emission. Therefore, stripping of hot galactic gas through both ram pressure and tidal forces does occur in groups and clusters, but the frequency or efficiency of such events must be moderate enough to allow hot gas halos in a large fraction of bright galaxies to survive even in group and cluster cores.
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