I have a set of $n\times n$ matrices with entries on $\mathbb{F}_2$, given by $$\mathcal{A}=\left\{A\in\mathcal{M}_{n\times n}(\mathbb{F}_2):A= \left( \begin{array}{ccc} I_{k_1} & 0 & 0 \\ 0 & J & 0 \\ 0 & 0 & I_{k_3} \end{array} \right) \right\} $$ where $J=\left( \begin{array}{cccc} 0 & \ldots & 0 & 1 \\ 0 & \ldots & 1 & 0\\ \vdots & \vdots & \vdots & \vdots\\ 1 & \ldots & 0 & 0\end{array} \right)_{b\times b}$ is a square matrix of size $b$ and $I_{k_1},I_{k_3}$ are identity matrices such that $k_1+k_3+b=n$
This set acts a flip block set for the column vectors $X$ such that $X^T=(x_1,x_2,\ldots,x_n)\in\mathbb{F}_2^n$.
I want to break the set of those vectors in some kind of equivalence classes such that every reversed vector $Y$ can be identified with just one of those equivalence classes.
For example, in the case of $n=5$ and $b=3$ the vectors with Hamming weight $1$ are divided in $2$ sets: $\left\{(1,0,0,0,0),(0,0,1,0,0),(0,0,0,0,1)\right\}$ and $\left\{(0,1,0,0,0),(0,0,0,1,0)\right\}$
Because no matter what $A\in\mathcal{A}$ you multiply any of these vectors they will stay in their sets.
I want to get this for any case but I don't know how to express it.