I want to prove the following fact:
If $G$ is a finite group scheme acting freely by $\mu$ on an abelian variety $X$ and $\pi \colon X \rightarrow X/G$ is the quotient map then for any coherent sheaf $\mathcal{F}$ on $X/G$ then there is a lift of the $G$-action to the pullback $\pi^{*}\mathcal{F}$.
I have read Mumford's proof in his book Abelian Varieties but I am struggling to prove the details he omits. Let $\mathcal{G} := \pi^{*}\mathcal{F}$. There are natural isomorphisms $\lambda_{1}:p_{X}^{*}\mathcal{G} \rightarrow (\pi \circ p_{X})^{*}\mathcal{F}$ and $\lambda_{2}:\mu^{*}\mathcal{G} \rightarrow (\pi \circ \mu)^{*}\mathcal{F}$. Using the fact that $\pi \circ p_{X} = \pi \circ \mu$ because $\pi$ is $G$-invariant, we can define an isomorphism $p_{X}^{*}\mathcal{G} \rightarrow \mu^{*}\mathcal{G}$ as $\lambda_{2}^{-1} \circ \lambda_{1}$. Now Mumford says to use the functorial properties of pullbacks to prove that this is in fact a lift of the $G$ action. This amounts to showing that a diagram is commutative. I am struggling with this because we are not working up to isomorphism, we are working with commutative diagrams where all the morphisms are isomorphisms.
It seems like an important example of descent so I thought I should try and work through the details; is this a bad idea? If not then is there a simple way of showing that this is in fact the lift of $\mu$?
