Let $\pi: Y \to X$ be a Galois cyclic cover with automorphism group $G$ generated via $g$, that arises from a line bundle $L$ on $X$ that is $n$ torsion.
I will give lots of context, but my question is how to characterize the image $\pi_* D^b(Y) \to D^b(X)$
Context
Given an identification $ \pi^*(L) = \mathcal{O} $ we obtain a canonical isomorphism $$\pi_*(\mathcal{O}) = \oplus^n L^i$$
Now since $\pi$ is affine we easily have that $\pi_* Quasi(Y) \subset Quasi(X)$ are determined via giving a $\pi_*(\mathcal{O})$ module structure.
The question is given some $\mathcal{F} \in Quasi(X)$, when can we enrich it to have such action. For $F$ a bundle, we in fact must have an isomorphism $i: F \cong F \otimes L$.
When $F$ is simple (i.e $End(F) = \mathbb{C}$), this is sufficient. Indeed in this case we can enrich the isomorphism to an action of the algebra $\pi_*(\mathcal{O})$. The point is that from any isomorphism $i$ gives an isomorphism $i: F=L^n \otimes F \cong F$ (the first using $L^n = \mathcal{O}$ which we want to be trivial. If it's not, we can fix it via choosing an nth root of it and changing $i$ appropriately.
A paper I'm looking at claims that on the derived category, the assumption that the sheaf is simple is not important; see section 2.2 of this- i.e $F \in D^b(X)$ is a pushforward iff $F = F \otimes L$
The fact $F$ is not a bundle is not very important, since in the derived world we resolve with bundles, and assuming each is a pushforward, we can probably get the morphisms to be compatible with the action. What's more surprising to me is that simplicity is not an issue, shouldn't we require the isomorphism $L \otimes F = F$ to be compatible with $L^n \otimes F = F$?