  The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of
the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal
subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at
studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the
automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns out
that any such automorphism leaves $\mathcal{O}_2$ globally invariant.
Furthermore, the subgroup ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shown
to be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. Saying exactly what the
group is amounts to understanding when an automorphism of $\mathcal{O}_2$ that
fixes $\mathcal{D}_2$ pointwise extends to $\mathcal{Q}_2$. A complete answer
is given for all localized automorphisms: these will extend if and only if they
are the composition of a localized inner automorphism with a gauge
automorphism.
