Let $F_2=\{0,1\}$ and consider a cyclic group of order $n$ acting on $F_2^n$ by cyclically exchanging the canonical basis vectors. Which are the orbits of such an action?
Let us see it for $n=4$. The cyclic group $G$ of order $4$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}:=\{[0],[1],[2],[3]\}$. If I take the canonical basis vectors of $F_2^4$, I have $(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)$ as canonical basis. Now, let $(1,1,0,0)=(1,0,0,0)+(0,1,0,0)$. The action of $G$ on $(1,1,0,0)$ is as follows:
$$ 0 \cdot (1,1,0,0)=(1,1,0,0) $$
$$ 1 \cdot (1,1,0,0)=(0,1,0,0)+(0,0,1,0)=(0,1,1,0) $$
and so on?