The setup seems like it must be this:
Then G induces an action on the power set of X, . Each subset of X corresponds to some graph with verticies V. Each permutation of the verticies (obviously ) represents a graph isomophism, and visa-versa, so that the orbit under G of a subset A of X (i.e. of a graph) is the set of graphs isomorphic to A.
There's a complication to consider in that and represent the same edge, so we could make them equivalent, and so consider only X/~, and let G act on . That does seem like a sensible way to proceed, as otherwise we'll be wrongly treating and as different graphs, where . I haven't thought it out, but that seems like how it should be handled. Without the equivalence classes, we'd be considering digraphs, not graphs.
That sets it up to apply the CFB Lemma. So the problem becomes computing when . I haven't thought about that, but this maybe will get you started.