  In a seminal paper, Brin demonstrates that the outerautomorphism group of
Thompson group $T$ is isomorphic to the cyclic group of order two. In this
article, building on characterisation of automorphisms of the Higman-Thompson
groups $G_{n,r}$ and $T_{n,r}$ as groups of transducers, we give a new proof,
automata theoretic in nature, of Brin's result.
  We also demonstrate that the group of outerautomorphisms of Thompson's group
$V = G_{2,1}$ contains an isomorphic copy of Thompson's group $F$. This extends
a result of the author demonstrating that whenever $n \ge 3$ and $1 \le r < n$
the outerautomorphism groups of $G_{n,r}$ and $T_{n,r}$ contain an isomorphic
copy of $F$.
