Name of subset of $X$ consisting of those elements $x$ fixed by a given subgroup $F \leq G$.

Question

Suppose that a group $G$ acts on a set $X$. Given $x \in X$, denote by $G_x$ the stabiliser subgroup of $G$ with respect to $x$, i.e. the subgroup of $G$ comprising all $g \in G$ which fix $x$. Now suppose that $F$ is a subgroup of $G$, and consider the set $\{x \in X : F \text{ is the stabiliser subgroup of } G \text{ with respect to } x\}$. What is the name of this set ?

Answer 1

In french we sometimes use $Fix_g = \{x\in X : g(x) = x\}$

but I advise you against using it as it is sometimes used in english to mean

$G_T = Fix_G(T) = \{g\in G : g(x) = x, \forall x\in T\}$

I am unaware if any other notations exist.