Suppose I have a scheme $X$ over $\mathbb C$, acted on by a finite group $G$. Let $\mathcal F$ be a $G$-equivariant coherent sheaf of $\mathcal O_X$-modules. Then I can form the stack quotient $[X/G]$, and $\mathcal F$ descends to $\mathcal F'$ on this stack.
Suppose that I know the dimensions of the groups $H^i([X/G], \mathcal F')$. What can I say about $\operatorname{dim} H^i(X, \mathcal F)$?
I know that $(H^i(X, \mathcal F))^G \cong H^i([X/G], \mathcal F')$, but ideally I'd like to know if there is something like an upper bound for $\operatorname{dim} H^i(X, \mathcal F)$. If there is no answer in such generality, I'd be happy to know of special cases.
The simplest example showing that no bound is possible is the following. Let $X = \mathrm{Spec}(\Bbbk)$ be a point, and let $F_1$ be the equivariant sheaf that corresponds to a non-trivial irreducible representation $V$ of $G$. Then $$ H^0(X,F_1) = V, \qquad\text{but}\qquad H^0([X/G],F_1') = V^G = 0. $$ Now if you take $F_n = F_1^{\oplus n}$, the dimension of $H^0(X,F_n)$ grows with $n$ to infinity, while $H^0([X/G],F_n') = 0$ for all $n$.