We confirm the detection of three groups in the Lynx supercluster, at z approximate to 1.3 through spectroscopic follow-up and X-ray imaging, and we give estimates for their redshifts and masses. We study the properties of the group galaxies compared to the two central clusters, RX J0849+4452 and RX J0848+4453. Using spectroscopic follow-up and multi-wavelength photometric redshifts, we select 89 galaxies in the clusters, of which 41 are spectroscopically confirmed, and 74 galaxies in the groups, of which 25 are spectroscopically confirmed. We morphologically classify galaxies by visual inspection, noting that our early-type galaxy (ETG) sample would have been contaminated at the 30%-40% level by simple automated classification methods (e.g., based on Sersic index). In luminosity-selected samples, both clusters and groups show high fractions of bulge-dominated galaxies with a diffuse component that we visually identified as a disk and which we classified as bulge-dominated spirals, e.g., Sas. The ETG fractions never rise above approximate to 50% in the clusters, which is low compared to the fractions observed in other massive clusters at z approximate to 1. In the groups, ETG fractions never exceed approximate to 25%>. However, overall bulge-dominated galaxy fractions (ETG plus Sas) are similar to those observed for ETGs in clusters at z similar to 1. Bulge-dominated galaxies visually classified as spirals might also be ETGs with tidal features or merger remnants. They are mainly red and passive, and span a large range in luminosity. Their star formation seems to have been quenched before experiencing a morphological transformation. Because their fractions is smaller at lower redshifts, they might be the spiral population that evolves into ETGs. For mass-selected samples of galaxies with masses M > 10(10.6) M-circle dot within Sigma > 500 Mpc(-2,) the ETG and overall bulge-dominated galaxy fractions show no significant evolution with respect to local clusters, suggesting that morphological transformations might occur at lower masses and densities. The ETG mass-size relation shows evolution toward smaller sizes at higher redshift in both cluster and groups, while the late-type mass-size relation matches that observed locally. When compared to the clusters, the group ETG red sequence shows lower zero points (at similar to 2 sigma) and large scatters, both expected to be an indication of a younger galaxy population. However, we show that any allowed difference between the age in groups and clusters would be small when compared to the difference in age in galaxies of different masses.