Comparisons modulo a prime for the number of $(0,1)$-matrices

Let $B$ be a set of $(0,1)$ $n\times n$ matrices such that if $M\in B$ and $M'$ is obtained from $M$ by an arbitrary rearrangement of rows and columns, then $M'\in B$. For prime $p$ we find comparisons modulo $p$ for $|B|$, where $|B|$ is the number of elements in $B$. We consider applications of this result in cases when $B$ is 1) a set of matrices with a permanent equal to $r$, $r\in\mathbb N_0=\{0,1,2,\cdots\}$; 2) a set of matrices with given row and column sums.