I am studying Ascbacher's Finite Group Theory, Second Edition (Cambridge University Press). I am trying to understand Lemma 5.2 which states:
If $H \unlhd G$ [where G is a group] then G acts on $\operatorname{Fix}(H)$. More generally G permutes the orbits of cardinality $c$, for each $c$.
I can prove the first statement by following the relevant definitions. However, I do not understand the second statement. What is this part saying?
Definitions
Notation
$G$ is a group of permutations on the set $X$. For $x \in X$, $xg$ denotes the image of $x$ under the permutation $g \in G$. (This is the author's notation for what is more commonly written as $g(x)$ these days.)
Group acting on a set
Let $Y$ be a subset of [a set] $X$. $G$ 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 $g \in G$.
Fixed points
For $S \subseteq G$ define
$$\operatorname{Fix}(S) = \{x \in X : xs = x, \forall s \in S\}$$
For $x \in \operatorname{Fix}(H)$, the singleton $\{x\}$ is an orbit of cardinality $1$ of the action of $H$. The first statemant says that the action of $G$ permutes these orbits of cardinality $1$.
The second statement then says that if $xH$ is an orbit of cardinality $c$ of $H$, then $xHg$ is another such orbit of cardinality $c$. Furthermore, it says that the map thus induced on the set of orbits of cardinality $c$ is a permutation.