Let $G$ be a compact group (possibly finite) acting holomorphically on a complex manifold $(M, J)$. Then we have an averaging map $$\rho: \Omega^{p,q}(M) \rightarrow \Omega^{p,q}(M)$$ $$\omega \mapsto \int_{G} g^*\omega ~~ dg$$
where $\int_{G} dg =1$.
Is it true that $\rho \circ \overline{\partial} (\omega) = \overline{\partial} \circ \rho(\omega)$ for all $\omega \in \Omega^{p,q}(M)$?