Let $g \geq 2$ be an integer and consider the symmetric group $S_n$ where $n = 2g+1$ or $n = 2g+2$ as a subgroup of the symplectic group $\mathrm{Sp}_{2g}(\mathbb{F}_2)$ via the standard representation(s). (See this question for a description of standard representations of symmetric groups.) This induces the adjoint action of $S_n$ on the Lie algebra $\mathfrak{sp}_{2g}(\mathbb{F}_2)$ (which is an $\mathbb{F}_2$-vector space of dimension $2g^2 + g$). Can anyone tell me the dimension of the first group cohomology $H^1(S_n, \mathfrak{sp}_{2g}(\mathbb{F}_2))$? In particular, I want to know if this first cohomology vanishes (I expect the answer might be different for $n = 2g+1$ and $n = 2g+2$).

EDIT: Actually, I see that what I really need to understand for my purposes is the first group cohomology of $S_{2g+1}$ or $S_{2g+2}$ with coefficients in $\mathbb{F}_2^{2g}$ via the standard representation(s) mentioned above. I'm not sure if this is a simpler or more difficult thing to solve, but for the moment I don't know how to solve it except by attempting brute force. Of course I'm still curious about my initial question.

I see that some similar questions have been asked here which sort of skirt around this, but most of the literature referenced in the answers seems to address either much more complicated questions or simply cohomology with coefficients in $\mathbb{F}_2$. It would take me a while to extract what I need from it as my primary background is not in Lie theory.