I recently found an interesting statement on a forum. If two permutation matrices are conjugate in $GL(n,2)$ then the corresponding permutations are conjugate in $S_n$. You can find it here. Is this statement true for groups of permutation matrices which are conjugate in $GL(n,2)$.
Let $G,H$ are conjugate groups of permutation matrices in $GL(n,2)$. Are the corresponding groups conjugate in $S_n$?
A good question, the answer is no. Examples can be found with GAP or Magma.
The smallest example appears for $n = 8$. Consider the following two subgroups of $S_8$ isomorphic to $S_3$:
$$H = \langle\ (3,5)(4,7)(6,8),\ (3,6,7)(4,8,5)\ \rangle$$
$$K = \langle\ (1, 2)(5, 7)(6, 8),\ (1, 3, 2)(4, 6, 8)\ \rangle$$
Here $H$ has orbits of sizes $1$, $1$, $6$. Meanwhile $K$ has orbits of sizes $2$, $3$, $3$.
Thus $H$ and $K$ are not conjugate in $S_8$. But a computer calculation shows that the corresponding subgroups of $GL(8,2)$ are conjugate.
In fact the groups of permutation matrices corresponding to $H$ and $K$ are conjugate in $GL(8,K)$ for every field $K$.
For $\operatorname{char} K = 3$ you can also check this by a computation in $GL(8,3)$. In the case where $\operatorname{char} K \neq 2,3$; you can simply see this by computing the permutation character (use Maschke's theorem). Indeed for the permutation character of $X \leq S_n$, for $g \in X$ the character value $\chi(g)$ is the number of fixed points of $g$ on $\{1,\ldots,n\}$.
Thus in the example above you see immediately that $H$ and $K$ have the same permutation character, thus the corresponding permutation matrix groups are conjugate in characteristics $\neq 2,3$.
I note as another example that the assumption that the permutation characters are the same is not sufficient. For example, you can find $X \cong Y \cong C_2 \times C_2$ in $S_6$ such that $X$ and $Y$ have the same permutation character, but the matrix groups are not conjugate in $GL(6,2)$. (Of course the matrix groups are conjugate in characteristic $\neq 2$.)
