Can the matrix transpose be represented by $X^T = AXB$ for a given $A$ and $B$? Can the matrix transpose be represented by $X^T = AXB$ for a given $A$ and $B$?
I think it is possible, but please correct me if I am wrong--please see my attempt below. 

My thinking (not sure whether I am on the right path) is that one could employ $\mbox{vec}$ and Kronecker product operator. However, I am not sure whether this formally proves it, does it?
If I vectorize both sides, i.e., 
\begin{align}
X^T &= AXB \\
\Downarrow \\
\mbox{vec}\left( X^T \right) &= \mbox{vec}\left( AXB \right) \\
\mbox{vec}\left( X^T \right) &= \underbrace{\left(B^T \otimes A\right)}_{\equiv \ \  T_{m,n}}\mbox{vec}\left( X \right)  \\
\mbox{vec}\left( X^T \right) &= T_{m,n}\mbox{vec}\left( X \right),  \\
\end{align}
where $T_{m,n}$ is some permutation matrix (cf. http://www4.ncsu.edu/~pfackler/MatCalc.pdf). 
Similar question: Solve $AXB=X^\top$
 A: No, there are no such matrices $A$ and $B$.
If there were, we would have $\operatorname{Id}=\operatorname{Id}^T=A\operatorname{Id}B=AB$ and therefore $B=A^{-1}$.
Now, is there an invertible $2\times2$ matrix $A=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ such that$$\begin{pmatrix}0&1\\0&0\end{pmatrix}=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}?$$It is easy to see that the only such matrices are those of the type $\left(\begin{smallmatrix}0&1/c\\c&0\end{smallmatrix}\right)$.
However,$$\begin{pmatrix}0&1/c\\c&0\end{pmatrix}\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&1/c\\c&0\end{pmatrix}^{-1}=\begin{pmatrix}0&0\\0&1\end{pmatrix}\neq\begin{pmatrix}1&0\\0&0\end{pmatrix}^T.$$
A: I partially answer this question based on your thoughts and the comment given by @greg .
$\mbox{vec}(X^T)=T_{m,n} \mbox{vec}(X)$
Note that $T_{m,n}$ can be written as the following way (cf. https://en.wikipedia.org/wiki/Commutation_matrix $K^{(r,m)}$)
$T_{m,n}=\displaystyle\sum^m_{i=1}\displaystyle\sum^n_{j=1}J^{ij}_{mn}\otimes (J^{ij}_{mn})^T$, where $(J^{ij}_{mn})^T=J^{ji}_{nm}$.
$J^{ij}_{mn}$ is the single-entry matrix (cf. https://en.wikipedia.org/wiki/Single-entry_matrix), with the matrix size $m\times n$. Hence,
\begin{align}
\mbox{vec}(X^T) &= \left(\displaystyle\sum^m_{i=1}\displaystyle\sum^n_{j=1}J^{ij}_{mn}\otimes (J^{ij}_{mn})^T\right) \mbox{vec}(X)\\
&= \displaystyle\sum^m_{i=1}\displaystyle\sum^n_{j=1}\left(J^{ij}_{mn}\otimes J^{ji}_{nm}\right) \mbox{vec}(X)\\
&= \displaystyle\sum^m_{i=1}\displaystyle\sum^n_{j=1}\mbox{vec}\left( J^{ji}_{nm} X (J^{ij}_{mn})^T \right)\\
&= \mbox{vec}\left( \displaystyle\sum^m_{i=1}\displaystyle\sum^n_{j=1} J^{ji}_{nm} X J^{ji}_{nm} \right)
\end{align}
Therefore, we have
\begin{align}
X^T= \displaystyle\sum^m_{i=1}\displaystyle\sum^n_{j=1} J^{ji}_{nm} X J^{ji}_{nm}
\end{align}
You can obtain this relationship via directly using the definition of single-entry matrix as well. And if you could do some tranformation to remove two summations, then we can completely answer your question. Hope it helps.  
