  [Abridged] To understand the role of the galaxy group environment on galaxy
evolution, we present a study of radio luminosity functions (RLFs) of group
galaxies based on the VLA-COSMOS 3 GHz Large Project. The radio-selected sample
of 7826 COSMOS galaxies with robust optical/near-infrared counterparts,
excellent photometric coverage, and the COSMOS X-ray galaxy groups (M_200c >
10^13.5 M_sun) enables us to construct the RLF of group galaxies (GGs) and
their contribution to the total RLF since z ~ 2.3. Using the Markov chain Monte
Carlo algorithm, we fit a redshift-dependent pure luminosity evolution model
and a linear and power-law model to the luminosity functions. We compare it
with past RLF studies from VLA-COSMOS on individual populations of
radio-selected star-forming galaxies (SFGs) and galaxies hosting active
galactic nuclei (AGN). These populations are classified based on the presence
or absence of a radio excess concerning the star-formation rates derived from
the infrared emission. We find that the density of radio galaxies in groups is
low compared to the field at z ~ 2 down to z ~ 1.25, followed by a sharp
increase at z ~ 1 by a factor of 6, and then a smooth decline towards low
redshifts. This trend is caused by both decrease in the volume abundance of
massive groups at high-z and the changes in the halo occupation of radio AGN,
which are found by other studies to reside at smaller halo mass groups. This
indicates that the bulk of high-$z$ log10(M_200c/M_sun) > 13.5 groups must have
been forming recently, and so the cooling has not been established as yet. The
slope of the GG RLF is flatter compared to the field, with excess at high radio
luminosities. The evolution in the GG RLF is driven mainly by satellite
galaxies in groups. A drop in occurrence of AGN in groups at z > 1 by a factor
of 6, manifests an important detail on the processes governing galaxy
evolution.
