We present a multi-wavelength analysis of the properties of extremely red galaxy (ERG) populations, selected in the GOODS-South/Chandra Deep Field South field. By using all the photometric and spectroscopic information available on large deep samples of extremely red objects (EROs; 645 sources), IRAC EROs (IEROs; 294 sources), and distant red galaxies (DRGs; 350 sources), we derive redshift distributions, identify active galactic nucleus (AGN)-powered and star formation (SF)-powered galaxies, and, using the radio observations of this field, estimate robust (AGN-and dust-unbiased) SF rate densities ((rho) over dot(*)) for these populations. We also investigate the properties of "pure" (galaxies that conform to only one of the three ERG criteria considered) and "combined" (galaxies that verify all three criteria) sub-populations. Overall, a large number of AGNs are identified (up to similar to 30%, based on X-rays and mid-infrared criteria), the majority of which are type-2 (obscured) objects. Among ERGs with no evidence for AGN activity, we identify sub-populations covering a wide range of average SF rates, from below 10 M-circle dot yr(-1) to as high as 200 M-circle dot yr(-1). Applying a redshift separation (1 <= z < 2 and 2 <= z <= 3), we find significant evolution (an increase of a factor of 2 or higher) of. * for EROs and DRGs, while none is observed for IEROs. The former populations can contribute more than 20% to the global (rho) over dot(*) at 2 <= z <= 3. The emission from AGN activity is typically not strong in the ERG population, with AGNs increasing the average radio luminosity of ERG sub-populations by, nominally, less than 20%. AGNs are common, however, and, if no discrimination is attempted, this could significantly increase the. * estimate (by over 100% in some cases). Thus, and while the contribution of star-forming processes to the radio luminosity in galaxies with AGN remains uncertain, a comprehensive identification of AGNs in these populations is necessary to obtain meaningful results.
