An abelian group $G$ with ${\rm Aut}(G)$ non-abelian

Question

Could you give me a simple example of $G$ abelian with ${\rm Aut}(G)$ non-abelian? Otherwise how could I prove that $G$ abelian implies ${\rm Aut}(G)$ abelian. (I don't really think that's true)

Answer 1

There are also examples of finite abelian groups with non-abelian automorphism group, e.g., $$ \operatorname{Aut}(C_2\times C_2)\cong S_3. $$ This seems to be the easiest example.

Answer 2

Take $G=\mathbb{Z}^2$ its automorphism group $Gl(2,\mathbb{Z})$ is not commutative.