Large restricted sumsets in general Abelian groups

Let A, B and S be subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A∧SB={a+b:a∈A,b∈Banda−b∉S}. Let LS=maxz∈G|{(x,y):x,y∈G,x+y=zandx−y∈S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+LS implies A∧SB=G. We then prove that if |A|+|B|=|G|+LS then |A∧SB|≥|G|−2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+LS and |A∧SB|=|G|−2|S|. Moreover, in this case, we also provide the structure of the set G∖(A∧SB).