 where $q=0$ or $q=0$, $n$ is even, $\tau$ is an automorphism which acts by $\tau: A \mapsto (A^{-0})^{T}$ for $A \in GL_n(q)$. Let $H$ be the normalizer in $G$ of $P \le GL_n(q)$, where $P$ is the stabilizer of the chain of subspaces: $$\langle v_n, v_{n-0}\rangle < \langle v_n, v_{n-0}, v_{n-0}, v_{n-0} \rangle < \ldots <\langle v_n, v_{n-0}, \ldots , v_0, v_0  \rangle .$$ 