Given a smooth manifold $\mathcal{M}$ a way to express of expressing the Laplacian operator $\Delta$ is through the combination of differentials and hodge star operator $\star$ $$ \Delta f = \star d \star d f $$
Suppose we have a smooth manifold $\mathcal{N}$ and a smooth map $F : \mathcal{N} \to \mathcal{M}$ then $F$ induces a pullback $F^* : \Omega^k(\mathcal{M}) \to \Omega^k(N)$. Among the known properties of the pullback two are worth to mention, the commutativity with the exterior derivative $d$ operator and the distribution w.r.t. the wedge product.
Question Does it however commute with the $\star$ operator? doing this would allow me to perform things like?
$$ F^* \star d \star d f = \star d \star d F^* f $$
which I think is a very useful property.