A question on a possible cyclic sieving phenomenon?

Question

(This is an old MSE question from me, which did not get any answer, and when looking back seems interesting to post it here:)

Let $G$ be a finite group. Consider the set $X_G:=\cup_{H\le G} G/H$, where the disjoint union is taken over all left cosets of the subgroups $H \le G$. Let $S \subset G$ be a generating set for $G$ and $|g| :=$ word length with respect to $S$. For a left coset $gH$ define $|gH| := \min_{ h\in H} |gh|$. Define $\pi(G,t) = \sum_{x \in X_G} t^{|x|}$. Then $\pi(G,1) = |X_G|$. For $G =\mathbb{Z}/(n) \equiv C_n$ the cyclic group and $S=\{+1\}$, it seems that

$\pi(G,-1) = $ number of odd divisors of $n$ (OEIS: A001227). (1)

For $G =\mathbb{Z}/(n) \equiv C_n$ the cyclic group and $S=\{\pm1\}$, it seems that

$\pi(G,-1) = $ #(of divisors of $n$ of the form $4m+1$)$-$#(of divisors of $n$ of the form $4m+3$), (OEIS: A002654).

So this $\pi(G,t)$ is a polynomial which when inserting the $2$-th roots of unity, counts something. My conjecture is, that $(X_G, C_2, \pi(G,t))$ exhibits the cyclic sieving phenomenon, but for this I need a group action from the cyclic group $C_2$ to $X_G$ such that (1) is fullfilled:

$\pi(C_n,-1) = |X_{C_n}^{-1}| = $ number of odd divisors of $n$.

However, I tried $c*gH = g^{-1}H$ for $C_2 = \{1,c\}$ but this action, although natural, does not give the desired property, and I do not have a polynomial for this action to get a cyclic sieving phenomenon.

So my question is:

1. For the cyclic group $G:=C_n$, how does one define an action from $C_2$ to $X_{C_n}$ such that (1) is fullfilled? You do not have to prove that (1) is fullfilled, numerical coincidences should be ok for the first.
2. If there is such an action in 1), is it possible to define this for an arbitrary finite group $G$, hence $C_2$ acts on $X_G$?

Thanks for your help!

Answer 1

Here is a sketch for the case of the cyclic group and $S$ containing only its generator.

Let $m$ be odd and $n=2^\ell m$. Then $ (-1)\ast g^k H = \{ g^m e^{-1} | e \in g^k H \} $ defines an action of $C_2=\{+1, -1\}$ on the set of cosets of subgroups of the cyclic group $\langle g\rangle$ of order $n$:

$$ (-1)\ast\{ g^m e^{-1} | e \in g^k H \} = \{ g^m (g^m e^{-1})^{-1} | e \in g^k H \}. $$

The fixed points of this action are the cosets of the form $\langle g^k\rangle$ with $m = j k$:

$$ (-1)\ast\{g^{i k} | 0\leq i < n/k \} = \{g^{j k - i k} | 0\leq i < n/k \} = \{g^{(j-i) k} | 0\leq i < n/k \}. $$