Computational complexity of sizes and number of orbits of a group acting on a set Given a group action of a group $G$ on a set $X$, is there any way to relate the number of orbits, i.e. $|X/G|=|\{\{g\cdot x:g\in G\}:x\in X\}|$, to the sizes of the orbits, i.e. $|\{g\cdot x:g\in G\}|$ for $x\in X$?

I'm in particular interested in the relation between the computational complexity of these two questions. To be more precise, for a fixed group action, if I know that it's $\#P$-Complete to find the size of an orbit (given an element in $X$), can I then make any statement of the computational complexity of counting the number of orbits?

Note: I've now asked the same question on mathoverflow, since after two weeks there was only an related answer here.
Consider the following question: Given two elements of X, are they in the same orbit under the action of G? This problem is Graph Isomorphism complete.
Consider the following question: Given two elements of X, are they in the same orbit under the action of G? This problem is Graph Isomorphism complete.