  We study the Fourier--Stieltjes algebra of Roelcke precompact,
non-archimedean, Polish groups and give a model-theoretic description of the
Hilbert compactification of these groups. We characterize the family of such
groups whose Fourier--Stieltjes algebra is dense in the algebra of weakly
almost periodic functions: those are exactly the automorphism groups of
$\aleph_0$-stable, $\aleph_0$-categorical structures. This analysis is then
extended to all semitopological semigroup compactifications $S$ of such a
group: $S$ is Hilbert-representable if and only if it is an inverse semigroup.
We also show that every factor of the Hilbert compactification is
Hilbert-representable.
