Let $F$ be a combinatorial species. The exponential generating series of $F$ is defined to be $$ \sum_{n \geq 0} \frac{ \lvert F_n \lvert x^n}{n!} $$ It was observed by Baez and Dolan in their paper "From finite sets to Feymann diagrams" that the exponential generating series can be categorified to a 1-groupoid $$ \sum_{n \geq 0} F_n \times X^n // S_n $$ You recover the exponential generating series by taking groupoid cardinality and setting $ x = \lvert X \lvert $. The cycle index series is another important series attached to a combinatorial species. It is defined by $$ Z_F(x_1,x_2,\dots) = \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in S_n} \lvert {\rm Fix}(F[\sigma])\lvert x_1^{\sigma_1} x_2^{\sigma_2} \dots $$
Question: Is it possible to construct an $n$-groupoid whose groupoid cardinality is $Z_F(x_1,x_2,\dots)$?
By $n$-groupoid, I mean a homotopy $n$-type?