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The 2-adic ring  C *-algebraQ    2naturally contains a copy of the Cuntz algebraO    2and,  a fortiori , also of its diagonal subalgebraD    2with Cantor spectrum. This paper is aimed at studying the groupAutD    2(    Q    2    )of the automorphisms ofQ    2fixingD    2pointwise. It turns out that any such automorphism leavesO    2globally invariant. Furthermore, the subgroupAutD    2(    Q    2    )is shown to be maximal abelian inAut   (    Q    2    ). Saying exactly what the group is amounts to understanding when an automorphism ofO    2that fixesD    2pointwise extends toQ    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.

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