Isomorphism from $\mathrm{Aut}(\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z})$ to $S_3$

Notice that $\mathbb{Z}/ 2 \mathbb{Z} \times \mathbb{Z}/ 2 \mathbb{Z}$ is a two dimensional vector space over $\mathbb{Z}/ 2 \mathbb{Z}$. There are $3$ nonzero vectors $(1,0), (0,1), (1,1)$. Now, any automorphism $\phi$ of this vector space will be a linear transformation and so is determined by where it map $(1,0)$ and $(0,1)$. Moreover, we observe that if we are given the action of $\phi$ on one of these basis vectors, the image under $\phi$ of the remaining basis vector can be either of the other two nonzero vectors. Thus, we see that the automorphisms are precisely the permuations of the three nonzero vectors in this vector space.