TITLE: $G$- invariant part of push forward
QUESTION [7 upvotes]: Let $C$ be a smooth projective curve. Let $G$ be a finite group which acts on $C$. Let $C'=C/G$ the quotient of the action, which is a smooth curve. Then $f:C\rightarrow C'$ is a finite, possibly ramified morphism.
1) Let $L$ be a line bundle on $C$ which admits a $G$-action. What is $(p_*L)^{G}$? Suppose $L=p^*L'$ for some line bundle $L'$, then $(p_*L)^{G}=L'$. 
2) Suppose $L$ does not come from below, is $(p_*L)^G$=0?
Do we have a characterization as to when $(p_*L)^G=0$?
I would like to understand what happens both when $f$ is unramified and $f$ is ramified.
Edit: 
It looks like $p^*(p_*L)^G\subset L$. Also that $(p_*L)$ and $(p_*L)^G$ are locally free. By the above inclusion, $(p_*L)^G$ can be at most of rank one. 
3) Consider the quotient $p_*L/(p_*L)^G$, is that locally free as well? 
It would be great if someone can direct me to a reference where such things are explained.

REPLY [2 votes]: I assume you suppose the action of $G$ to be faithful. If the action is free, a structure of $G$-equivariant line bundle on $L$ is the same as a descent data for $$C \to C':= C/G : L \text{ is the pullback of } L':=(p_*L)^G.$$
I would need to think more when $C \to C'$ is not tame. Assuming tame, I like to think of $L$ as a line bundle on $C'$, viewed as an orbifold. One will have $L'$ is a line bundle on $C'$ and $L$ is $(p^*L')(D)$, where $D$ is on the ramification locus and: for each $x'$ in $C'$ above which $C$ ramifies with ramification $d(x')$ there is $n(x')$,$$0 \le n(x') < d(x')$$such that at $x$ above $x'$, $D$ is $n(x')x$. Note that as such a divisor $D$ is stable by $G$, a $L$ so constructed is clearly $G$-equivariant. Given $L'$, all the $L$ obtained from different $D$ as above have the same $(p_*)^G$. In the orbifold language, as a line bundle on the orbifold $C'$, $L$ is$$L'\left(\sum n(x')/d(x').x'\right).$$