Evolution of the anti-truncated stellar profiles of S0 galaxies since
  $z=0.6$ in the SHARDS survey: II - Structural and photometric evolution
  Type-III S0 galaxies present tight scaling relations between their surface
brightness photometric and structural parameters. Several evolutionary models
have been proposed for the formation of Type-III S0 galaxies but the
observations of are usually limited to the local Universe.
  We study the evolution of the photometric and structural scaling relations
found between the parameters of the surface brightness profiles in the
rest-frame R-band of Type-III S0 galaxies with z and the possible differences
between the rest-frame (B-R) colours of the inner and outer disc profiles.
  We make use of a sample of 14 Type-III E/S0--S0 galaxies at 0.2<z<0.6 to
study if the correlations found in local Type-III S0 galaxies were present ~6
Gyr ago. We analyse the distribution of the surface brightness characteristic
parameters as a function of the stellar mass and if there is a significant
change with z. We also derive their rest-frame (B-R) colour profiles and we
compare these results with the predictions from a grid of SSP models.
  We find that the inner and outer scale-lengths of Type-III S0 galaxies at
0.4<z<0.6 follow compatible trends and scaling relations with those observed in
local S0 galaxies. We do not detect any significant differences between the
location of Rbreak between z~0.6 and z=0 for a fixed stellar mass of the
object, whereas the surface brightness at the break radius is ~1.5 mag arcsec-2
dimmer in the local Universe than at z~0.6 for a fixed stellar mass.
  In contrast to Type-II profiles, the Type-III surface brightness profiles of
S0 galaxies present compatible Rbreak values and scaling relations during the
last 6 Gyr. This result and the similarity of the colours of the inner and
outer discs point to a highly scalable and stable formation process, probably
more related to gravitational and dynamical processes than to the evolution of
stellar populations (abridged).
