Ribes and Zalesskii Corollary 4.4.8 show that the group of continuous automorphisms of $\widehat{\mathbb{Z}}$ satisfies ${\rm Aut}\,(\widehat{\mathbb{Z}})\cong\mathbb{Z}_2\times\frac{\mathbb{Z}}{2\mathbb{Z}}\times\prod\limits_{p\in\mathbb{P}}[\mathbb{Z}_p\times\frac{\mathbb{Z}}{(p-1)\mathbb{Z}}]$.

Let $F$ be a rank-$2$ dense free abelian subgroup of $\widehat{\mathbb{Z}}$. Which continuous automorphisms of $\widehat{\mathbb{Z}}$ restrict to a $\mathbb{Z}$-linear automorphism of $F$? Does the answer depend on the algebraicity of a basis of $F$?


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