Quotient of profinite completion of the integers by the integers

Definition
This group is defined as the quotient group $$\hat{\mathbb{Z}}/\mathbb{Z}$$, where $$\hat{\mathbb{Z}}$$ is the defining ingredient::profinite completion of the integers and $$\mathbb{Z}$$ is the group of integers sitting inside $$\hat{\mathbb{Z}}$$ in the obvious way.