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We consider a class of groups $V_n(G)$ which are supergroups of theHigman-Thompson groups $V_n$. These groups fit in a framework of ElizabethScott for generating infinite virtually simple groups, and the groups we studyin 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 treeautomorphism 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 specificvertex $\alpha$ of the infinite rooted $n$-ary tree according to thepermutation they represent, and then they iterate this permutation again ateach 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 furthernon-isomorphism results in the other cases. We show the perhaps surprisingresult that if $G$ does act freely, then $V_n(G)\cong V_n$. We also generalisethese results and produce some examples of even more isomorphisms amongstgroups in the family $V_n(G)$.  Essential tools in the above work are a study of the dynamics of the actionof elements of $V_n(G)$ on Cantor space, Rubin's Theorem, and transducers fromGrigorchuk, Nekrashevych, and Suschanski\u{i}'s rational group on the $n$-aryalphabet.

Some isomorphism results for Thompson like groups $V_n(G)$