Context : Anti-truncated lenticular galaxies ( Type-III S0s ) present tight scaling relations between their surface brightness photometric and structural parameters .
Although several evolutionary models have been proposed for the formation of these structures , the observations of Type-III S0 galaxies are usually limited to the local Universe .

Aims : We aim to compare the properties of Type-III discs in a sample of S0 galaxies at 0.2 < z < 0.6 with those of the local Universe .
In this paper , we study the evolution of the photometric and structural scaling relations measured in the rest-frame R - band with z and the possible differences between the rest-frame ( B - R ) colours of the inner and outer disc profiles .

Methods : We make use of a sample of 14 Type-III E/S0–S0 galaxies at 0.2 < z < 0.6 from the GOODS-N field identified and characterised in a previous paper .
We study whether or not the correlations found in local Type-III S0 galaxies were present \sim 6 Gyr ago .
We analyse the distribution of the surface brightness characteristic parameters ( R _ { \mathrm { break } } , \mu _ { \mathrm { break } } , h _ { \mathrm { i } } , h _ { \mathrm { o } } , \mu _ { 0 , \mathrm { i } } and \mu _ { 0 , \mathrm { o } } ) as a function of the stellar mass and look to see if there is a significant change with z .
We also derive their rest-frame ( B - R ) colour profiles .
Finally , we compare these results with the predictions from a grid of SSP models .

Results : 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 as a function of the break radius , R _ { \mathrm { break } } .
We do not detect any significant differences between the location of R _ { \mathrm { break } } between z \sim 0.6 and z \sim 0 for a fixed stellar mass of the object , whereas the surface brightness at the break radius \mu _ { \mathrm { break } } is \sim 1.5 mag arcsec ^ { \mathrm { -2 } } dimmer in the local Universe than at z \sim 0.6 for a fixed stellar mass .
We find no significant differences in the ( B - R ) colour between the inner and outer profiles of the Type-III S0 galaxies at 0.2 < z < 0.6 .

Conclusions : In contrast to Type-II ( down-bending ) profiles , the anti-truncated surface brightness profiles of S0 galaxies present compatible R _ { \mathrm { break } } 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 .