Context: Consider, two $m \times n$ matrices $A, B$ such that there is a permutation $\kappa$ that such that such that $A^{\kappa}=B$ (Wielandt's notation), i.e. $A, B$ are isomorphic but not equal.
Since, permutations of rows are independent of permutations of columns, so $\kappa = \sigma \pi$ where $\sigma$ is a row permutation that acts on rows of non-symmetric matrix and $\pi$ acts on columns of matrix.
A set of column-permutations $\beta_k$ is a subset of the set of all possible permutations of columns of $A$. So, $\beta_k \subset S_n$.
A set of column-permutations $\gamma_k$ is a subset of the set of all possible permutations of columns of $B$. So, $\gamma_k \subset S_n$.
We consider 2 different set due to different labeling of columns in $A, B$ .
$\pi$ is in the set $\beta_k, \gamma_k$ but with different label.
This is equivalent to bipartite graph isomorphism.
$\textit{Unique Column :}$ If in a matrix, there exists one column which has no similar column in that matrix, regardless any row permutation , then that column is an Unique Column.
$\textit{Example:}$ In the matrix $ \mathcal{A}$, $2^{nd}$ column is unique. In the $ \mathcal{B}$, all three columns are unique. $$\mathcal{A}=\left( \begin{array}{ccc} 1 & 0 & 1\\0 & 0 & 0\\0 & 1 & 0\\ \end{array} \right) , \mathcal{B}=\left( \begin{array}{ccc} 1 & 0 & 0 \\0 & 1 & 0\\0 & 0 & 1\\ \end{array} \right) $$
Obseravation:
Though we have different labeling for $\pi$ in $\beta_k, \gamma_k$, $\pi$ in $\beta_k$ will change the position of some unique columns of $A$ to the same position exactly as $\pi$ in $\gamma_k$ will do in $B$.
For example,if $\pi$ in $\beta_k$ changes the column of $1^{st}$ position to the column of $6^{th}$ positionof $A$ then $\pi$ in $\gamma_k$ will change the column of $1^{st}$ position to the column of $6^{th}$ position in $B$.
Thus we can find/distinguish $\pi$ or an automorphism of $A^{\pi}$in both $\beta_k, \gamma_k$.
Question:
Is this observation correct all the time? Is there an exception?