I am not sure if I should ask this as a separate question, but I was also wondering the following. The partition function of (untwisted) Dijkgraaf Witten theory for a finite group G just counts the number of principal bundles up to gauge transformations. However, when we have DW theory with defects, how do we count principal bundles with prescribed boundaries and defects?In arxiv.org/abs/1507.00941, they proposed a topological invariant for the case of defects but they are missing a representation theory formula that generalizes the ones above. Is it somewhere in the literature?

Thank you for your answer. Actually my question exactly came from studying 2d TQFT. However, I couldn't find Proportion 2.13 in your nice paper. Do you maybe mean Theorem 1.4, which has a similar expression for the symmetric group?

How does your formula for $Z(\Sigma)$ generalize when $\Sigma$ has $n$ boundaries with specified holonomies $k_i$? Actually I asked this question here mathoverflow.net/questions/371707/…, and I just saw your post.