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The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy ofthe Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonalsubalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed atstudying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of theautomorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns outthat any such automorphism leaves $\mathcal{O}_2$ globally invariant.Furthermore, the subgroup ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shownto be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. Saying exactly what thegroup is amounts to understanding when an automorphism of $\mathcal{O}_2$ thatfixes $\mathcal{D}_2$ pointwise extends to $\mathcal{Q}_2$. A complete answeris given for all localized automorphisms: these will extend if and only if theyare the composition of a localized inner automorphism with a gaugeautomorphism.

Diagonal automorphisms of the $2$-adic ring $C^*$-algebra