Non trivial homomorphisms $\mathbb{Z}/3\mathbb{Z} \rightarrow \text{Aut}(\mathbb{Z}/7\mathbb{Z})$

Question

I have to find non trivial homomorphisms $\varphi$, $\varphi'$: $\mathbb{Z}/3\mathbb{Z} \rightarrow \text{Aut}(\mathbb{Z}/7\mathbb{Z})$. We know that Aut $(\mathbb{Z}/7\mathbb{Z}) \cong (\mathbb{Z}/7\mathbb{Z})^\times \cong \mathbb{Z}/6\mathbb{Z}$. So we have to look order of elements of $\mathbb{Z}/6\mathbb{Z}$ that divide $3$. These elements are $[2]_6$ and $[3]_6$. In corrections these homomorphisms are defined as: $\Big(\varphi([1]_3)\Big)([i]_7) = [2i]_7$ and $\varphi'$:$\Big(\varphi([1]_3)\Big)([i]_7) = [4i]_7$. The thing that i don't understand, is what $[i]_7$ represents. For me, if we generalize for example $\varphi$, we got $\varphi([r]_3) = [2^r]_7$. I read a lot of articles on ''research'' of homomorphisms, and i feel confused right now and can't get an intuition for this. Thanks in advance for help.