Does this property characterizes amenability or there are examples of non-amenable groups satisfying it?

Let $G$ be finitely generated group.

Property:

There exists $C<1$ such that for every $S\subset G$ - finite set, there exists $F \subset G$ - finite, such that

$|sF \Delta F|\leq C\cdot |F|$ for every $s\in S$

Currently I don't see if it is possible to rebuilt sets $F$ in order to construct Folner sequence for $G$.

The question is related to Amenability of groups