I understand your explanation. But there seem only to be 12 such pairs.We can assume the free abelian group of rank 2 is [tex]\mathbb{Z}\times\mathbb{Z}[/tex].
The homomorphisms from [tex]\mathbb{Z}\times\mathbb{Z}[/tex] to [tex]S_3[/tex], correspond to pairs [tex](a,b)\in S_3\times S_3[/tex] with [tex]ab=ba[/tex] via
[tex]f:\mathbb{Z}\times\mathbb{Z}\to S_3[/tex] corresponds to [tex](f(1,0),f(0,1))[/tex].
(Show this!) So you just need to find all such pairs.
Tip for enumerating these pairs: There are only three types of elements in [tex]S_3[/tex]: the identity, the transpositions and the cyclic permutations.
I see now. thank you.I found 18 pairs. Did you count the pairs (a,b), (b,a) as the same? They give different homomorphisms.