  We performed a detailed study of the evolution of the star formation rate
(SFR) and stellar mass of the brightest group galaxies (BGGs) and their
relative contribution to the total baryon budget within $R_{200}$
($f^{BGG}_{b,200}$). The sample comprises 407 BGGs selected from X-ray galaxy
groups ($M_{200}=10^{12.8}-10^{14} \;M_{\odot}$) out to $z\sim1.3$ identified
in the COSMOS, XMM-LSS, and AEGIS fields. We find that BGGs constitute two
distinct populations of quiescent and star-forming galaxies and their mean SFR
is $\sim2$ dex higher than the median SFR at $ z<1.3 $. Both the mean and the
median SFRs decline with time by $>2$ dex. The mean (median) of stellar mass of
BGGs has grown by $0.3$ dex since $z=1.3$ to the present day. We show that up
to $\sim45\% $ of the stellar mass growth in a star-forming BGG can be due to
its star-formation activity. With respect to $f^{BGG}_{b,200}$, we find it to
increase with decreasing redshift by $\sim0.35$ dex while decreasing with halo
mass in a redshift dependent manner. We show that the slope of the relation
between $f^{BGG}_{b,200}$ and halo mass increases negatively with decreasing
redshift. This trend is driven by an insufficient star-formation in BGGs,
compared to the halo growth rate. We separately show the BGGs with the 20\%
highest $f^{BGG}_{b,200}$ are generally non-star-forming galaxies and grow in
mass by processes not related to star formation (e.g., dry mergers and tidal
striping). We present the $ M_\star-M_h $ and $ M_\star/M_h-M_h $ relations and
compare them with semi-analytic model predictions and a number of results from
the literature. We quantify the intrinsic scatter in stellar mass of BGGs at
fixed halo mass ($\sigma_{log M_{\star}}$) and find that $\sigma_{log
M_{\star}}$ increases from 0.3 dex at $ z\sim0.2 $ to 0.5 dex at $ z\sim1.0 $
due to the bimodal distribution of stellar mass.
