Let $G$ be a permutation group acting on some set. Let $C(g)$ be the set of associated cycles of an element $g\in G$, and define $l(c)$ to be the length of a cycle $c$. Now set
$$T(G) = \sum_{g\in G}\sum_{c\in C(g)}x^{l(c)}\\ = \sum_{h} \left|\textrm{Cl}(h)\right|\sum_{c\in C(h)}x^{l(c)} ,$$
where the sum over $h$ is over representatives of conjugacy classes $\textrm{Cl}(h)$.
Is there a name/prior research on this polynomial? I am in particular interested in a 'pointed version', i.e. $\overline{T}(G) = x \partial_x T(G)$. For the cylic groups (and their natural action), $\overline{T}$ is the reciprocal polynomial of the cycle index polynomial. For the symmetric groups (and their action on $n$ objects), one recovers a polynomial proportional to a finite geometric series.
I am also interested in `twisted' versions of $T$:
$$T(G,\chi) = \sum_{g\in G}\chi(g)\sum_{c\in C(g)}x^{l(c)}\\ = \sum_{h}\left(\chi(h) \left|\textrm{Cl}(h)\right|\right)\sum_{c\in C(h)}x^{l(c)},$$
for $\chi$ a character of $G$.
