I'm interested in the relation between the computational complexity of counting orbits and counting elements in orbits for groups acting on sets. More formally:

Assume that $X_n$ is a infinite sequence of finite sets index by $n\in\mathbb{N}$. Assume that $G$ is a group with a finite set of generators. Furthermore assume that a group action of $G$ is defined on each set $X_n$ and that this action can be efficiently computed given an element of $G$ and an element of $X$.

Consider the following two problems:

**Problem 1 (counting orbits):** Input: $n$

Given $n$, decide the number of orbits of $G$ acting on $X_n$, i.e. compute the size $|X_n/G|=|\{\{g\cdot x:g\in G\}:x\in X_n\}|$.

**Problem 2 (size of orbit):** Input: $x\in X_n$

Given $x\in X_n$, decide the number of elements in the orbit of $G$ acting on $X_n$ containing x, i.e. compute the size $|\{g\cdot x:g\in G\}|$.

Does $\#P$-Completeness of **Problem 2** imply $\#P$-Completeness/Hardness of **Problem 1**?

Note that $\#P$-Completeness of **Problem 2** implies that the size of $X_n$ must scale super-polynomially with $n$.

*Note: I've asked a similar question on StackExchange/Mathematics, however after two weeks and only an unrelated answer I thought I'd also ask the same question here.*

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