We use Chandra data to study the incidence and properties of active galactic nuclei (AGNs) in 16 intermediate redshift (0.5 < z < 1.1) X-ray-selected galaxy groups in the Chandra Deep Field-South. We measure an AGN fraction of at , approximately a factor of two higher than the AGN fraction found for rich clusters at comparable redshift. This extends the trend found at low redshift for groups to have higher AGN fractions than clusters. Our estimate of the AGN fraction is also more than a factor of three higher than that of low redshift X-ray-selected groups. Using optical spectra from various surveys, we also constrain the properties of emission-line selected AGNs in these groups. In contrast to the large population of X-ray AGNs (N(L X, H > 1041 erg s-1) = 25), we find only four emission-line AGNs, three of which are also X-ray bright. Furthermore, most of the X-ray AGNs in our groups are optically dull (i.e., lack strong emission-lines), similar to those found in low redshift X-ray groups and clusters of galaxies. This contrasts with the AGN population found in low redshift optically selected groups which are dominated by emission-line AGNs. The differences between the optically and X-ray-selected AGNs populations in groups are consistent with a scenario where most AGNs in the densest environments are currently in a low accretion state. © 2014. The American Astronomical Society. All rights reserved..