Consider the space $\mathbb{F}_2^n$, and a group $G$ acting on this space. I can then pick a point $x \in \mathbb{F}_2^n$, and obtain an orbit $G \cdot x = \{gx : g \in G\}$.
It is then possible to verify that $\dim(G\cdot x) = \dim(G) - \dim(G_x)$, with $G_x$ the stabilizers of $x$.
However, I would like to be able to quantify the dimension of $\mathbb{F}_2^n[G\cdot x]$ instead, i.e. the subspace generated by $G \cdot x$. Is there a general formalism that tackles this question?