Name of a set that arises in a group action: $S(x,y) = \{g \in G:\ gx = y\}$ for $x,y \in X$

Question

In the context of an action of a group $G$ on a set $X$, I was wondering if there is a name for the set $$S(x,y) = \{g \in G:\ gx = y\}$$ for $x,y \in X$. It's not mentioned on the Wikipedia or other resources that I checked.