Here $Sp(2n,\mathbb{F}_2)$ means the group of matrices preserving the form $\Omega = \left( \begin{array}{cc} 0&I \\ -I&0& \end{array} \right)$, i.e. the symplectic group over an even characteristic ($-1=1$). It is well known that, $Sp(2n,\mathbb{F}_2) \leq SL(2n,\mathbb{F}_2)$. Additionally both groups are generated by transvections, but for $Sp(2n,\mathbb{F}_2)$ the transvections need to preserve the form $\Omega$.

Say $S$ is a generating set such that $Sp(2n,\mathbb{F}_2) = \langle S \rangle$, and $X \in SL(2n,\mathbb{F}_2)$ but$X \notin Sp(2n,\mathbb{F}_2)$, then does $\langle S, X \rangle = SL(2n, \mathbb{F}_2)$ hold?

It's clear that $SL(2,\mathbb{F}_2) = Sp(2,\mathbb{F}_2)$, yet I'm not sure about the general case. It's tedious, but possible to show it for $n=2$ with elementary methods, but the induction seems unclear. I am thankful for suggestions.