On the size of sets avoiding a general structure

Given a finite abelian group $G$ and a subset $S\subseteq G$, we let $N_{G,\ S}$ be the smallest integer $N$ such that for any subset $A\subseteq G$ with $N$ elements, we have $g+S\subseteq A$ for some $g\in G$. Using the probabilistic method, we prove that \begin{align*} \frac{|H_G(S)|-1}{|H_G(S)|}|G|+\Biggl\lceil\biggl(\frac{|G|}{|H_G(S)|}\biggr)^{1-|H_G(S)|/|S|}\Biggr\rceil\le N_{G,\ S}\le \biggl\lfloor\frac{|S|-1}{|S|}|G|\biggr\rfloor+1, \end{align*} where $H_G(S)$ is the stabilizer of $S$.