I want to find all the group homomorphisms $\varphi:\mathbb{Z}/4\mathbb{Z} \to \text{Aut}(\mathbb{Z}/5\mathbb{Z})$.
I first want to know $\text{Aut}(\mathbb{Z}/5\mathbb{Z})$ and the orders of all its elements: $\text{Aut}(\mathbb{Z}/5\mathbb{Z})\cong (\mathbb{Z}/5\mathbb{Z})^{\times} \cong \mathbb{Z}/4\mathbb{Z}$, and as cyclic group automorphisms send generators to generators each $\sigma \in \text{Aut}(\mathbb{Z}/5\mathbb{Z})$ is the "multiplication by k" map $\sigma_k$ where $k \in \{1,2,3,4\}$ are the generators in $\mathbb{Z}/5\mathbb{Z}$. We also have the sanity check that each $|\sigma_k|$ must divide $|\text{Aut}(\mathbb{Z}/5\mathbb{Z})|=4$.
$\sigma_1(x)=\text{Id}_{\mathbb{Z}/5\mathbb{Z}}(x)=x$ has order 1.
$\sigma_2(x)=2x$ has order 4, right? Since we need to compose 4 times before we get $16x=x\mod5$.
$\sigma_3(x)=3x$ and $\sigma_4(x)=4x$ both have order 2 similarly. Please kindly verify if these are correct.
Now our $\varphi$: we know there are $4$ of them as $|\text{Hom}_{\mathbb{Z}}(\mathbb{Z}/4\mathbb{Z}, \text{Aut}(\mathbb{Z}/5\mathbb{Z}))|=4$, and our determining values of $\varphi(1)$ must have order dividing $|\mathbb{Z}/4\mathbb{Z}|=4$ and $|\text{Aut}(\mathbb{Z}/5\mathbb{Z})|=4$, so we can map $\varphi(1)$ to elements of order $1,2$ or $4$ in $\text{Aut}(\mathbb{Z}/5\mathbb{Z})$. Hence,
$\varphi(1)=\sigma_1$ (order 1), $\varphi(1)=\sigma_3$ and $\varphi(1)=\sigma_4$ (order 2), and $\varphi(1)=\sigma_2$ (order 4). Yes?
From here, I am unsure how to find the image of the elements other than 0 and 1. Attempt: for example let's try $\varphi(1)=\sigma_3$. This map is $0 \mapsto \sigma_1, 1 \mapsto \sigma_3, 2 \mapsto ?, 3 \mapsto ?$.
Is $\varphi(2)=\varphi(1+1)=\varphi(1)+\varphi(1)=\sigma_3 \circ \sigma_3=\sigma_1$? Do I continue in this way? Is there a quicker method I should be seeing?