# Subgroup of ${\rm Aut}\,(\widehat{\mathbb{Z}})$

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 $$F$$?