Find all homomorphisms $\varphi: C_4 \to V$ or $\psi: V \to C_4$ where $C_4$ is the cyclic group and $V$ the Kleinian group

An exercise of our first problem set in geometry is to find all homomorphisms $\varphi: C_4 \to V$ or $\psi: V \to C_4$, where $C_4$ is the cyclic group and $V$ the Kleinian group. I have no idea how to start with this exercise. Can anyone give me a hint?

-

If $g$ is a generator of $C_4$ then $\phi$ is completely given by specifying $\phi(g)\in V$; you must make sure that $\phi(g)^4=1$, but that is true for all elements of $V$. Hence you find four $\phi\colon C_4\to V$.
On the other hand, $\psi$ is given by specifying the values of the two generators. But thes values must be of order dividing 2, not arbitrary elements of $C_4$. There are two such elements, $1$ and $g^2$, hence you have $2\cdot 2$ homomorphisms $\psi\colon V\to C_4$.