Let $k$ be an algebraically closed field. Any finite $k$-morphism $P^1_k\rightarrow P^1_k$ is flat (miracle flatness) and surjective on the underlying spaces. Therefore, the pushforward of a coherent locally free sheaf is coherent locally free (on $P^1_k$, such sheaves can be described by a finite sequence of integers using the fact that Picard rank is 1 and there is Birkhoff--Grothendieck splitting).

Assume we have a finite $k$-morphism $P^1_k\rightarrow P^1_k$ such that the inverse image of the generic point has cardinality $n\geq 2$. Which sheaves can we get as the pushforward of a locally free sheaf of rank 1?