In Finite Group Theory, 2nd Edition, Aschbacher states
For $Y \subseteq X$, I'll sometimes write $C_G(Y)$ ... for $G_Y$.... Usually this notation will be used only when $X$ possesses a group structure preserved by $G$.
Why is this notation justified? Are $C_G(Y)$ and $G_Y$ equivalent in some way or under certain situations?
Notation
$G$ is a group of permutations on a set $X$. That is, $G \subseteq Sym(X)$.
Function application and composition are denoted as simple juxtaposition and associate from left to right, rather than the more modern right to left. For example, $xg$ denotes the image of $x \in X$ under the permutation $g \in G$. This is the author's notation for what we now typically write as $g(x)$. Similarly, $gf$ the composition of the permutation $f$ with the permutation $g$. In other words, it is $f \circ g$.
Definitions
Centralizer
For $X \subseteq G$...define $C_G(X) = \{g \in G : xg = gx, \forall x \in X\}$.
Group Action
Let $Y$ be a subset of $X$. $G$ is said to act on $Y$ if $Y$ is a union of orbits of $G$. Notice $G$ acts on $Y$ precisely when $yg \in Y$ for each $y \in Y$, and each $g \in G$.
Pointwise stabilizer
$G_Y = \left\{g \in G : yg = y, \forall y \in Y\right\}$
Discussion
A large part of my confusion here is that $C_G(X)$ is defined for a subset of the group $G$ whereas $G_Y$ is defined for a set which does not have a subset relationship with $G$. I take the phrase "when $X$ possesses a group structure preserved by $G$" to mean that the permutations of $G$ are homomorphisms of the group $X$. Is this correct? Does this allow us to think of $X$ as a subset of $G$ in some manner?
I guess the author by "X possesses a group structure preserved by G" means an action of $G$ that maps $X$ to itself. In other words the group action on $X$ induces and authomorphism on $X$.
A common example of this is when $X$ is a normal subgroup of $G$ and $G$ acts on it by conjugation. And indeed in this example we have that $G_Y = C_G(Y)$. But this relation isn't true when $G$ acts on $X$ by left multiplication.