  We study automorphism groups of randomizations of separable structures, with
focus on the $\aleph_0$-categorical case. We give a description of the
automorphism group of the Borel randomization in terms of the group of the
original structure. In the $\aleph_0$-categorical context, this provides a new
source of Roelcke precompact Polish groups, and we describe the associated
Roelcke compactifications. This allows us also to recover and generalize
preservation results of stable and NIP formulas previously established in the
literature, via a Banach-theoretic translation. Finally, we study and classify
the separable models of the theory of beautiful pairs of randomizations,
showing in particular that this theory is never $\aleph_0$-categorical (except
in basic cases).
