Examples of torsion-free abelian groups with finite automorphism group $\mathbb{Z}$ is a torsion-free abelian group with finite automorphism group. Are there other examples of such groups? 
Jumping from $\mathbb{Z}$ to $\mathbb{Q}$ is not good; since $\mathbb{Q}$ has infinite automorphism group. Is there any group in-between $\mathbb{Z}$ and $\mathbb{Q}$ with required property?

Exercise: If $f$ is any map from the set of primes to the set of non-negative integers, then (1) the subgroup $B_f$ of $\mathbf{Q}$ generated by $1/p^{f(p)}$ when $p$ ranges over primes has automorphism group reduced to $\{\pm 1\}$.
And (2) $B_f$ and $B_g$ are isomorphic if and only if $f,g$ coincide outside a finite subset. (In particular these are uncountably many non-isomorphic groups.) 
