Let $G$ be a finite group of measure-preserving measurable maps on $(\Omega, \mathcal{A}, \mathbf{P})$ and let $\mathcal{A}_{0}:=\{A \in \mathcal{A}: g(A)=A$ for all $g \in G\}$.
Show that, for every $X \in \mathcal{L}^{1}(\mathbf{P})$, we have
$$
\mathbf{E}\left[X \mid \mathcal{A}_{0}\right]=\frac{1}{\# G} \sum_{g \in G} X \circ g.
$$