Consider the generalization of the Rubik cube problem, where we are given an initial state $A$ and a final state $B$, and we search for a path between the two.
We can easily show that given $A$ and $B$, one can compute $X$ such that the path from $X$ to $S$ is the same as the path from $A$ to $B$. Here, $S$ denotes the standard solved state.
Now, is there a simple and convenient way to represent this isomorphism between the group subsumed by $A$ and $B$ and the group subsumed by $X$ and $S$?
Let $G_1$ be the group such that $A$ and $B$ are members of $G_1$. Accordingly, let $G_2$ be the group such that $X$ and $S$ are members of $G_2$. How to characterize the isomorphism between $G_1$ and $G_2$, which maps $A$ to $X$ and $B$ to $S$?
The isomorphism basically consists in recoloring the minicubes in $A$ to generate $X$ in such a way that the solution (path) from $X$ to $S$ is the same as the solution from $A$ to $B$. Note that we are able to compute $X$ without solving the Rubik cube problem, which is nice, because actually $X$ and $S$ may be in a different connected component than $A$ and $B$.
Correspondingly, we are searching for an isomorphism that does not require that $A$ and $B$ be in the same connected component as $X$ and $S$. Naturally, the isomorphism cannot rely on solving the Rubik cube, as the solution would forcefully require the four elements to be in the same connected component.
As an example, consider the following initial state $A$, depicted below,
The goal is to find a path to state $B$, below
This is equivalent to finding a path from state $X$ below to the canonical target state $S$
How to describe, using the language of group theory, the isomorphism that maps $A$ into $X$ and that maps $B$ into the canonical target state $S$?
In what follows, we focus on the simplest case wherein $A$, $B$, $X$ and $S$ are all in the same connected component. The possible configurations that we can get by manipulating a cube in configuration $A$ is the "orbit" of $A$. So, we're saying that in what follows the orbit of $A$ (which includes $B$) must be the same as the orbit of $S$ (which includes $X$).
We can think about each configuration of the cube as being an alias to a sequence of moves that brings that configuration to the canonical solved state $S$.
Say that we refer to $M1$ as the sequence of moves that brings $A$ to the solved state $S$, and we refer to $M2$ as the sequence of moves that brings $B$ to the solved state $S$.
Then, the sequence of moves that brings $A$ to $B$ is
$M1 M2^{-1}$
and $ A$ maps to $M1$ and $B$ maps to $M2$
in the same way that
$X$ maps to $M1 M2^{-1}$
and
$S$ maps to $M2 M2^{-1} $ which is the identity element.
Now, to build this mapping between $A$ and $X$ and between $B$ and $S$ we had to take the original sequence of moves and multiply each of those by $M2^{-1}$ (assuming operations are applied from left to right, otherwise, we would get $M2^{-1}M1$ as the mapping of $X$ instead of $M1 M2^{-1}$)
I'd be curious to know if there is an alternative way to represent that mapping, which does not rely on knowing the sequence of moves $M2^{-1}$. We can write a computer program to build $X$ from $A$ and $B$ without knowing $M1$ and $M2$, but I don't know how to represent the operator that this program is implementing using the language of group theory.
Is there an operator that maps states without necessarily sticking to the cube rules, but which allows to map $A$ into $X$ and $B$ into $S$ without knowing $M2$? I believe this operator is a sort of exogenous or "out of band" operator in the group elements, leveraging side information about alternative representations of the elements of the groups.