A vector of odd weight meets a transitive group (characteristic two)

Question

Let $C \subset \mathbb{F}_2^n$ be a $G$-invariant subspace, where $G \subset S_n$ acts transitively on the coordinates of $\mathbb{F}_2^n$. Assume that

1. $|G_i|$ is even for each coordinate $1 \leq i \leq n$,
2. $C$ contains a vector of odd Hamming weight less than $n$,
3. the weight $n$ vector does not belong to $C$.

For which odd $n$ such a pair $(C,G)$ exists?

Just curious about whether or not transitivity + 2 contradict to 3 when 1 holds. They obviously do when 1 does not hold.