We introduce our survey of galaxy groups at 0.85 < z < 1, as an extension of the Group Environment and Evolution Collaboration. Here we present the first results, based on Gemini GMOS-S nod-and-shuffle spectroscopy of seven galaxy groups selected from spectroscopically confirmed, extended XMM detections in COSMOS. We use photometric redshifts to select potential group members for spectroscopy, and target galaxies with r < 24.75. In total, we have over 100 confirmed group members, and four of the groups have >15 members. The dynamical mass estimates are in good agreement with the masses estimated from the X-ray luminosity, with most of the groups having 13 < log Mdyn/M⊙ < 14. We compute stellar masses by template-fitting the spectral energy distributions; our spectroscopic sample is statistically complete for all galaxies with Mstar≳ 1010.1 M⊙, and for blue galaxies we sample masses as low as Mstar∼ 108.8 M⊙. The fraction of total mass in galaxy starlight spans a range of 0.25–3 per cent, for the six groups with reliable mass determinations. Like lower redshift groups, these systems are dominated by red galaxies, at all stellar masses Mstar > 1010.1 M⊙. A few group galaxies inhabit the ‘blue cloud’ that dominates the surrounding field; instead, we find a large and possibly distinct population of galaxies with intermediate colours. The ‘green valley’ that exists at low redshift is instead well populated in these groups, containing ∼30 per cent of the galaxies. These do not appear to be exceptionally dusty galaxies, and about half show prominent Balmer absorption lines. Furthermore, their Hubble Space Telescope morphologies appear to be intermediate between those of red-sequence and blue-cloud galaxies of the same stellar mass. Unlike red-sequence galaxies, most of the green galaxies have a disc component, but one that is smaller and less structured than discs found in the blue cloud. We postulate that these are a transient population, migrating from the blue cloud to the red sequence, with a star formation rate that declines with an exponential time-scale 0.6 < τ < 2 Gyr. Such galaxies may not be exclusive to the group environment, as we find examples also amongst the non-members. However, their prominence among the group galaxy population, and the marked lack of blue, star-forming galaxies, provides evidence that the group environment either directly reduces star formation in member galaxies or at least prevents its rejuvenation during the normal cycle of galaxy evolution.