We have a commutative monoid M, and a function $f: M \rightarrow \{0, 1\}$. We're promised that this function factors through a (surjective) homomorphism to a finite abelian group (considered as a commutative monoid), $M \rightarrow G \rightarrow \{0, 1\}$, but not told what the group is.

Can we recover any information about the structure of G, just given $f$? I'm particularly interested in the case where M is the monoid of positive integers under multiplication. 

More generally, and more interestingly: Can we recover any information about the structure of G, *assuming that the black-box is allowed to lie about a small fraction of values of the function*?