I know that when finding homomorphisms between groups, for a cyclic group to any other group, then the homomorphism is completely determined by where you send the generator. However, I have two questions regarding homomorphisms between non-abelian groups and abelian groups.
For instance, I know that for a homomorphism between $S_3$ and $C_4$ (cyclic group of order 4), you map the commutator subgroup, $A_3$ to the unit element of $C_4$. However, what do you do with the even permutations, and why must you map the commutator subgroup to the unit element of $C_4$?
Also, how should I approach finding the homomorphisms between $C_2 \times C_2$ (direct product of two cyclic groups of order 2) and $S_3$?