No. of representatives of the elements of $Bw_{\sigma} B$ in $GL_3(\mathbb F_q)\setminus B$ (the set of all right cosets of $B$)

Question

Let $K=\mathbb F_q$. Let $B$ be the set of all upper triangular matrices in $GL_3(K)$. For $\sigma \in S_3$, let $w_\sigma := \begin{pmatrix} e_{\sigma(1)} & e_{\sigma(2)} & e_{\sigma(3)} \end{pmatrix}$, where $e_i$ is the $3\times 1$ column vector whose $i$-th entry is $1$ and the rest two entries are $0$.

For a given $\sigma \in S_3$, how to find the size of the following set of right cosets of $B$ : $\{B (\alpha w_\sigma \beta) : \alpha,\beta \in B\}$ ?