Question about methods of finding homomorphisms from $\mathbb Z_4$ to $\mathbb Z_2 \times \mathbb Z_2$
I've seen both methods, but one seems to fail here. What went wrong?
$\mathbb Z_4=\langle 1\rangle$, so homomorphism is defined by the value we give to $f(1)$. So we can give $4$ values, hence there are $4$ homomorphisms.
We know $\ker f\trianglelefteq \mathbb Z_4$, so $|\ker f|$ is either $1,2,4$.
With this method, I can only find $2$ homomorphisms, trivial and another in which $|\ker f|=2$. What went wrong?