How many distinct homomorphisms $\phi$ $:$ $C$$_3$ $\rightarrow$ $C$$_1$$_2$ are there? How many of these $\phi$ are ring homomorphisms $\Bbb Z$$_3$ $\rightarrow$ $\Bbb Z$$_1$$_2$?
Let $C$$_3$ $=$ $<x>$ and $C$$_1$$_2$ $=$ $<y>$. Then assignment $\rho$($x$) $=$ $y$$^k$ extends to a group homomorphism $\rho$($x$$^m$) $=$ $y$$^m$$^k$ iff $y$$^3$$^k$ $=$ $1$ iff $12$$\vert$$3k$ iff $4$$\vert$$k$. Thus, there are 3 distinct homomorphisms ($k$ $=$ $0,4,8$). This completes the first part.
How do I find the second part?
Hint: a ring homomorphism must send $1$ to $1$.
If your class$^*$ doesn't require ring homomorphisms to preserve the multiplicative identity, then at a minimum, $1$ must be sent to an idempotent $e$, i.e., $e^2=e$. Use the Chinese Remainder Theorem to break down $\mathbb Z_{12}$.
$*$ Or whatever context you might be studying this in.