How do we determine set of all automorphisms of an infinte cyclic group?

Question

Let $G=\langle a\rangle$ be an infinite cyclic group. $Aut(G)$ be set of all automorphisms of $G$. How to determine $Aut(G)$.

Answer 1

Hint 1: A homomorphism $h:G\to H$ is uniquely determined by where it sends the generators of $G$. This is also true for when $H = G$.

Hint 2: Automorphisms must be surjective.

Answer 2

Here $G \cong \mathbb{Z}$. You can show that any automorphism $\phi$ can send $1$ either to $1$ or $-1$ (otherwise $\phi$ will not be onto). Hence $Aut(\mathbb{Z}) \cong \mathbb{Z}_{2}$.