Automorphism from $\Bbb Q$ to $\Bbb Q$

Let $f:\mathbb{Q}\to \mathbb{Q}$ be such a homomorphism, and let $q = f(1)$, then

1. For any $m\in \mathbb{Z}$, $$ f(m) = mf(1) = mq $$
2. For any $r = m/n \in \mathbb{Q}$, you have $$ nf(r) = f(m) = mq \Rightarrow f(r) = q(m/n) = qr $$

If $1 \to q$, then

• $2 \to ?$
• $1/2 \to ?$