  The unique characteristics of the brightest group galaxies (BGGs) link the
evolutionary continuum between galaxies like the Milky Way and more massive
BCGs in dense clusters. This study investigates the stellar properties of BGGs
over cosmic time (z = 0.08-1.30), extending our previous work (Gozaliasl et al.
2016, 2018; Paper I and Paper II). We analyze data of 246 BGGs from our X-ray
galaxy group catalog in the COSMOS field, examining stellar age, mass, star
formation rate (SFR), specific SFR (sSFR), and halo mass. Comparisons are made
with Millennium and Magneticum simulations. We explore the variation of stellar
properties with the projected offset from the X-ray peak or host halo center.
Using a mock galaxy catalog, we evaluated the accuracy of SED-derived stellar
ages, finding a mean absolute error of about one Gyr. Observed BGG age
distributions show a bias towards younger ages compared to semi-analytical
models and the Magneticum simulation. Our analysis of stellar age versus mass
reveals trends with a positive slope, suggesting complex evolutionary pathways
across redshifts. We observe a negative correlation between stellar age and SFR
across all redshift ranges. Using a cosmic-time-dependent main sequence
framework, we identify star-forming BGGs, finding that about 20% of BGGs in the
local universe exhibit star-forming characteristics, increasing to 50% at
$z=1.0$. Our findings support an inside-out formation scenario for BGGs, where
older stellar populations are near the X-ray peak and younger populations at
larger offsets indicate ongoing star formation. The distribution of stellar
ages for lower-mass BGGs ($10^{10-11} M_\odot$) deviates from constant ages
predicted by models, highlighting current models' limitations in capturing
galaxies' complex star formation histories.
