  We aim to interpret important constructions in the theory of automorphisms of
the shift dynamical system in terms of subgroups $\mathcal{L}_{n,r}$ of the
outer-automorphism groups $\mathcal{O}_{n,r}$ of the Higman--Thompson group
$G_{n,r}$, and to extend results and techniques in
$\operatorname{Aut}(X_n^{\mathbb{Z}}, \sigma_{n})$ to the groups of
automorphisms $\operatorname{Aut}(G_{n,r})$ and outerautomrphisms of the
Higman--Thompson group $G_{n,r}$.
  Our mains results are a concrete realisation of the "inert subgroup",
important subgroup in the study of automorphism groups of shift spaces, as a
subgroup $\mathcal{K}_{n}$ of $\mathcal{L}_{n,n-1}$.
  Using this realisation, we show that the $\operatorname{Aut}(G_{n,r})$
contains an isomorphic copy of $\operatorname{Aut}(X_{m}^{\mathbb{Z}},
\sigma_{m})$ for all $m \ge 2$.
  A survey of the literature then yields that $\operatorname{Aut}(G_{n,r})$
contains isomorphic copies of finite groups, finitely generated abelian groups,
free groups, free products of finite groups, fundamental groups of 2-manifolds,
graph groups and countable locally finite residually finite groups to name a
few.
  We extend a result for $\operatorname{Aut}(X_n^{\mathbb{Z}}, \sigma_{n})$ to
the group $\mathcal{O}_{n,n-1}$. The homeomorphism
$\overleftarrow{\phantom{a}}$ of $X_n^{\mathbb{Z}}$ which maps a sequence
$(x_i)_{i \in \mathbb{Z}}$ to the sequence $(y_{i})_{i \in \mathbb{Z}}$ defined
such that $y_{i} = x_{-i}$ induces an automorphism
$\overleftarrow{\mathfrak{r}}$ of $\operatorname{Aut}(X_n^{\mathbb{Z}},
\sigma_{n})$, and consequently, an automorphism of $\mathcal{L}_{n}$. We extend
the automorphism $\overleftarrow{{\mathfrak{r}}}$ to the group
$\mathcal{O}_{n,n-1}$.
  In a forthcoming article, we demonstrate that the group $\mathcal{O}_{n}$ is
isomorphic to the mapping class group of the full two-sided shift over $n$
letters.
