Dimension of null space of symmetric binary matrix

Question

Let $A$ and $B$ be binary $n \times n$ matrices.

Edit: That is to say, $A,B \in M_n(GF(2))$ acting on the natural vector space $GF(2)^n$.

Suppose also that $B$ is symmetric.

What can be said about the dimension of the null space of $B+ABA^T$ (over $GF(2)$)?

For example is it nicely bounded above?

Thanks.

Edit: More specifically, what can be said if $B \neq 0$ and $A$ preserves no subspace of $GF(2)^n$