  We verify a conjecture of Vershik by showing that Hall's universal countable
locally finite group can be embedded as a dense subgroup in the isometry group
of the Urysohn space and in the automorphism group of the random graph. In
fact, we show the same for all automorphism groups of known infinite
ultraextensive spaces. These include, in addition, the isometry group of the
rational Urysohn space, the isometry group of the ultrametric Urysohn spaces,
and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$.
Furthermore, we show that finite group actions on finite metric spaces or
finite relational structures form a Fra\"iss\'{e} class, where Hall's group
appears as the acting group of the Fra\"iss\'{e} limit. We also embed continuum
many non-isomorphic countable universal locally finite groups into the isometry
groups of various Urysohn spaces, and show that all dense countable subgroups
of these groups are mixed identity free (MIF). Finally, we give a
characterization of the isomorphism type of the isometry group of the Urysohn
$\Delta$-metric spaces in terms of the distance value set $\Delta$.
