  We describe, through the use of Rubin's theorem, the automorphism groups of
the Higman-Thompson groups $G_{n,r}$ as groups of specific homeomorphisms of
Cantor spaces $\mathfrak{C}_{n,r}$. This continues a thread of research begun
by Brin, and extended later by Brin and Guzm\'an: to characterise the
automorphism groups of the `Chameleon groups of Richard Thompson,' as Brin
referred to them in 1996. The work here completes the first stage of that
twenty-year-old program, containing (amongst other things) a characterisation
of the automorphism group of $V$, which was the `last chameleon.' The
homeomorphisms which arise fit naturally into the framework of Grigorchuk,
Nekrashevich, and Suschanskii's rational group $\mathscr{R}$: they are exactly
those homeomorphisms which are induced by bi-sychronizing transducers, which we
define in the paper. This result appears to offer insight into the nature of
Brin and Guzman's exotic automorphisms, while also uncovering connections with
the theory of reset words for automata (arising in the Road Colouring Problem)
and with the theory of automorphism groups of the full shift.
