In abstract groups $\Gamma$ the normalizer $N_\Gamma(S)$ of a subset $S\subseteq\Gamma$ is the subgroup of all $x \in \Gamma$ that commute with $S$, i.e. $xS = Sx$, i.e. $x\ y\ x^{-1} \in S $ for all $y \in S$.
Among the permutations $S_n$ of the vertices of a graph $G$ of order $n$ (or any other kind of structure) there is one distinguished subgroup: the automorphisms $\text{Aut}(G)$, that reflect the symmetries of $G$:
$$\alpha \in \text{Aut}(G)\quad \Leftrightarrow \quad \alpha G = G$$
To give $\alpha G = G$ a proper meaning, identify $G$ with an adjacency matrix, for example.
The normalizer of $\text{Aut}(G)$ is another distinguished subgroup: it consists of those permutations $\pi$ of the vertices, such that $\text{Aut}(\pi G) = \text{Aut}(G)$, i.e. that respect the symmetries of $G$, as can be shown like this:
$\quad \pi \in N_{S_n}(\text{Aut}(G))\\ \Leftrightarrow \pi^{-1}\alpha\ \pi \in \text{Aut}(G)\ \text{for all}\ \alpha \in \text{Aut}(G)\\ \Leftrightarrow \pi^{-1}\alpha\ \pi\ G = G\ \text{for all}\ \alpha \in \text{Aut}(G)\\ \Leftrightarrow \alpha\ \pi\ G = \pi\ G\ \text{for all}\ \alpha \in \text{Aut}(G)\\ \Leftrightarrow\alpha \in \text{Aut}(\pi G)\ \text{for all}\ \alpha \in \text{Aut}(G) $
Note that $\text{Aut}(\pi G)$ and $ \text{Aut}(G)$ are of course isomorphic for every $\pi \in S_n$:
$$\text{Aut}(\pi G) \simeq \text{Aut}(G)$$
but this is not the matter of concern. The matter of concern is
$$\text{Aut}(\pi G) = \text{Aut}(G)$$
My first question now is:
Does the normalizer of the automorphisms of a structure has an established name on its own?
Something like symmetry preserving rearrangements (compared to adjaceny preserving rearrangements [what automorphisms are] or structure preserving rearrangements [what general permuations - of labels - are])? Note, that and how the following permutation is (i) symmetry preserving and (ii) an element of the normalizer of $\text{Aut}(G)$ and that (iii) most other permuations are not:
Where is the normalizer of the automorphisms of a structure investigated in its own right - or plays an explicit role, e.g. in a theorem?
More specific:
How can the normalizer of the automorphisms of a structure be defined/characterized without reference to (and prior definition of) the latter?