This follows up on what I thought was a good question which has now been deleted, asking about the automorphisms of the multiplicative group $\mathbf{Q}^{*}$.
Is there a relatively simple description, in group-theoretic terms, of the group of automorphisms of the $\mathbf{Z}$-module $\bigoplus_{n \in \mathbf{N}} \mathbf{Z}$?
And can such a description be given if instead we have $(\mathbf{Z}/2\mathbf{Z}) \oplus \bigoplus_{n \in \mathbf{N}} \mathbf{Z}$?
I see that the factor $\mathbf{Z}/2\mathbf{Z}$ would need to be preserved under any automorphism, since it is the torsion submodule.