Let $G$ and $H$ be two abelian groups. Is it true that $\text{Aut}(G\times H)\cong\text{Aut}(G)\times\text{Aut}(H)$?

Question

Let $G$ and $H$ be two abelian groups. Is it true that $\text{Aut}(G\times H)\cong\text{Aut}(G)\times\text{Aut}(H)$?

I am trying to figure out $\text{Aut}(G)$ where $G=\mathbb{Z}/5\mathbb{Z}\bigoplus \mathbb{Z}/25\mathbb{Z}$.

Answer 1

No, we have $\rm{Aut}(C_2\times C_2)\cong S_3$, see Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $D_6$, but ${\rm Aut}(C_2)$ is trivial.

Answer 2

If the groups are finite then it is true if $\gcd(|G|,|H|)=1$.