The niceness condition you want is on the action, not on the space $X$. Specifically, you want to have that $X\to X/G$ is a principle $G$-bundle, so that we have a Serre spectral sequence as you describefor $G\to X\to X/G$. Of course, since you're assuming that $G$ is a finite discrete group, the singular cohomology of $G$ is free, and only in degree 0. In fact, the requirement of being a "principle $G$-bundle" is the same as $X\to X/G$ being a covering space.
The problem: though the spectral sequence looks simple, and collapses immediately at the $E_2$ page, it's not really very useful, since all the interesting data is hidden in the local coefficient system (which is absolutely not trivial unless $G=0$.)
However, we can perhaps get the relationship you want in a much easier way if you're willing to modify the coefficient ring a bit. In particular, the answer is much simpler if you use a ring in which the order of $G$ is a unit. In that case, it's not hard to show directly (using covering space theory) that
$H^*(X/G)\to H^*(X)$ is an isomorphism onto the invariants of $G$, i.e. the subring
$H^*(X)^G$ of classes which are invariant under the action of $G$. This is an exercise in Milnor's Characteristic Classes, and I believe some form of it appears in Hatcher as well.