  We consider three extensions of the Navarro, Frenk and White (NFW) profile
and investigate the intrinsic degeneracies among the density profile parameters
on the gravitational lensing effect of satellite galaxies on highly magnified
Einstein rings. In particular, we find that the gravitational imaging technique
can be used to exclude specific regions of the considered parameter space, and
therefore, models that predict a large number of satellites in those regions.
By comparing the lensing degeneracy with the intrinsic density profile
degeneracies, we show that theoretical predictions based on fits that are
dominated by the density profile at larger radii may significantly over- or
underestimate the number of satellites that are detectable with gravitational
lensing. Finally, using the previously reported detection of a satellite in the
gravitational lens system JVAS B1938+666 as an example, we derive for this
detected satellite values of r_max and v_max that are, for each considered
profile, consistent within 1sigma with the parameters found for the luminous
dwarf satellites of the Milky Way and with a mass density slope gamma < 1.6. We
also find that the mass of the satellite within the Einstein radius as measured
using gravitational lensing is stable against assumptions on the substructure
profile. In the future thanks to the increased angular resolution of very long
baseline interferometry at radio wavelengths and of the E-ELT in the optical we
will be able to set tighter constraints on the number of allowed substructure
profiles.
