Find all group homomorphisms $\left(\mathbb{Z}/3\mathbb{Z} \right) ^\times\to \left(\mathbb{Z}/5\mathbb{Z}\right )^\times$

Question

Are there other group homomorphisms $f: \left(\mathbb{Z}/3\mathbb{Z} \right) ^\times\to \left(\mathbb{Z}/5\mathbb{Z}\right )^\times$ other than

$$f: \left(\mathbb{Z}/3\mathbb{Z} \right) ^\times\to \left(\mathbb{Z}/5\mathbb{Z}\right )^\times, ~x \mapsto e$$

with $e$ being the identity element – and if yes, how can I find them?

Answer 1

$1$ must be mapped to $1$.

$2^2=1$ in $(\mathbb Z/3\mathbb Z)^\times$,

so $2$ must be mapped to $1$ (which is the solution you mentioned) or $4$ in $(\mathbb Z/5\mathbb Z)^\times$.