  We prove that an arbitrary compact metrizable group can be realized as the
automorphism group of a graphing; this is a continuous analogue to Frucht's
theorem recovering arbitrary finite groups are automorphism groups of finite
graphs.
  The paper also contains a number of results the persistence of transitivity
of a compact-group action upon passing to a limit of graphons. Call a compact
group $\mathbb{G}$ graphon-rigid if, whenever it acts transitively on each
member $\Gamma_n$ of a convergent sequence of graphons, it also acts
transitively on the limit $\lim_n \Gamma$. We show that for a compact Lie group
$\mathbb{G}$ graphon rigidity is equivalent to the identity component
$\mathbb{G}_0$ being semisimple; as a partial converse to a result of
Lov\'{a}sz and Szegedy, this is also equivalent to weak randomness: the
property that the group have only finitely many irreducible representations in
each dimension. Similarly, call a compact group $\mathbb{G}$ image-rigid if for
every compact Lie group $\mathbb{H}$ the images of morphisms $\mathbb{G}\to
\mathbb{H}$ form a closed set (of closed subgroups, in the natural topology).
We prove that graphon rigidity implies image rigidity for compact groups that
are either connected or profinite, and the two conditions are equivalent (and
also equivalent to being torsion) for profinite abelian groups.
