  We consider a class of groups $V_n(G)$ which are supergroups of the
Higman-Thompson groups $V_n$. These groups fit in a framework of Elizabeth
Scott for generating infinite virtually simple groups, and the groups we study
in particular are initially introduced by Farley and Hughes. The group $V_n(G)$
is the result one obtains by taking the $V_n$ generators and adding a tree
automorphism for each generator of a subgroup $G$ of the symmetric group on $n$
letters, where the new generators each permute the child leaves of a specific
vertex $\alpha$ of the infinite rooted $n$-ary tree according to the
permutation they represent, and then they iterate this permutation again at
each vertex which is a descendent of $\alpha$.
  Farley and Hughes show that $V_n(G)$ is not isomorphic to $V_n$ when $G$
fails to act freely on the points $\{1,2,...,n\}$, and expect further
non-isomorphism results in the other cases. We show the perhaps surprising
result that if $G$ does act freely, then $V_n(G)\cong V_n$. We also generalise
these results and produce some examples of even more isomorphisms amongst
groups in the family $V_n(G)$.
  Essential tools in the above work are a study of the dynamics of the action
of elements of $V_n(G)$ on Cantor space, Rubin's Theorem, and transducers from
Grigorchuk, Nekrashevych, and Suschanski\u{i}'s rational group on the $n$-ary
alphabet.
