What are all the characteristic subgroups of $(\mathbb Q,+)$?

Question

What are all the characteristic subgroups of $(\mathbb Q,+)$? As in, the subgroups $H$ such that $f(H)=H, \forall f \in \operatorname{Aut}(\mathbb Q)$? I have reduced the condition to $qH=H$, $\forall q \in \mathbb Q\setminus \{0\}$.

Answer 1

Suppose $h \in H$ is non-zero. Let $g \in \mathbb{Q} \setminus \{0\}$. Then with $q=g/h$ and your reduction you get $g \in H$. Thus, if you have a non-zero element in $H$, then $H$ must be the full group.

This leaves only two possible subgroups.

Answer 2

For all $q \in \mathbb{Q} - \{0\}$, we must have $q H = H$, that is, for any element $h \in H$, we must have $q h \in H$.

If $h = 0$, the requirement is that $0 \in H$, which holds automatically because it is the identity of $\mathbb{Q}$.

What can you conclude if $H$ contains a nonzero element?