On the generalized restricted sumsets in abelian groups

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\}$$ contains at least $$\min\{|A|+|B|-3|S|,p(G)\}$$ elements, where $p(G)$ is the least prime factor of $|G|$. Further, we also have $$|A\stackrel{S}+B|\geq \min\{|A|+|B|-|S|-2,p(G)\},$$ provided that both $|A|$ and $|B|$ are large with respect to $|S|$.