  [ABRIDGED] We recently presented evidence of a connection between the
brightness profiles of nearby early-type galaxies and the properties of the AGN
they host. The radio loudness of the AGN appears to be univocally related to
the host's brightness profile: radio-loud nuclei are only hosted by ``core''
galaxies while radio-quiet AGN are only found in ``power-law'' galaxies. We
extend our analysis here to a sample of 42 nearby (V < 7000 km/s) Seyfert
galaxies hosted by early-type galaxies. We used the available HST images to
study their brightness profiles. Having excluded complex and highly nucleated
galaxies, in the remaining 16 objects the brightness profiles can be
successfully modeled with a Nuker law with a steep nuclear cusp characteristic
of ``power-law'' galaxies (with logarithmic slope 0.51 - 1.07). This result is
what is expected for these radio-quiet AGN based on our previous findings, thus
extending the validity of the connection between brightness profile and radio
loudness to AGN of a far higher luminosity. We explored the robustness of this
result against a different choice of the analytic form for the brightness
profiles, using a Sersic law. In no object could we find evidence of a central
light deficit with respect to a pure Sersic model, the defining feature of
``core'' galaxies in this modeling framework. We conclude that, regardless of
the modeling strategy, the dichotomy of AGN radio loudness can be univocally
related to the host's brightness profile. Our general results can be re-phrased
as ``radio-loud nuclei are hosted by core galaxies, while radio-quiet AGN are
found in non-core galaxies''.
