  Central Galaxies (CGs) in massive halos live in unique environments with
formation histories closely linked to that of the host halo. In local clusters
they have larger sizes ($R_e$) and lower velocity dispersions (sigma) at fixed
stellar mass M_star, and much larger R_e at a fixed $\sigma$ than field and
satellite galaxies (non-CGs). Using spectroscopic observations of group
galaxies selected from the COSMOS survey, we compare the dynamical scaling
relations of early-type CGs and non-CGs at z~0.6, to distinguish possible
mechanisms that produce the required evolution. CGs are systematically offset
towards larger R_e at fixed $\sigma$ compared to non-CGs with similar M_star.
The CG R_e-M_star relation also shows differences, primarily driven by a
sub-population (~15%) of galaxies with large $R_e$, while the M_star-sigma
relations are indistinguishable. These results are accentuated when double
Sersic profiles, which better fit light in the outer regions of galaxies, are
adopted. They suggest that even group-scale CGs can develop extended components
by these redshifts that can increase total $R_e$ and M_star estimates by
factors of ~2. To probe the evolutionary link between our sample and cluster
CGs, we also analyze two cluster samples at z~0.6 and z~0. We find similar
results for the more massive halos at comparable z, but much more distinct CG
scaling relations at low-z. Thus, the rapid, late-time accretion of outer
components, perhaps via the stripping and accretion of satellites, would appear
to be a key feature that distinguishes the evolutionary history of CGs.
