We investigate the galaxy populations in seven X-ray-selected, intermediate-redshift groups (0.2 < z < 0.6). Overall, the galaxy populations in these systems are similar to those in clusters at the same redshift; they have large fractions of early-type galaxies (f_e ~ 70%) and small fractions of galaxies with significant star formation (f[O II] ~ 30%). We do not observe a strong evolution in the galaxy populations from those seen in X-ray-luminous groups at low redshift. Both f_e and f[O II] are correlated with radius but do not reach the field value out to ~r₅₀₀. However, we find significant variation in the galaxy populations between groups, with some groups having fieldlike populations. Comparisons between the morphological and spectral properties of group galaxies reveal both gas-poor mergers and a population of passive spirals. Unlike low-redshift, X-ray-emitting groups, in some of these groups the brightest galaxy does not lie at the center of the X-ray emission, and in several of the groups that do have a central BGG, the BGG has multiple components. These groups appear to represent a range of evolutionary stages in the formation of the BGG. Some groups have relatively large central galaxy densities, and one group contains a string of seven bright galaxies within a radius of 200 kpc that have a lower velocity dispersion than the rest of the system. None of the central galaxies, including those with multiple components, have significant [O II] emission. These observations support a scenario in which BGGs are formed relatively late through gas-poor mergers.