  Using a result of Longo and Xu, we show that the anomaly arising from a
cyclic permutation orbifold of order 3 of a holomorphic conformal net $\mathcal
A$ with central charge $c=8k$ depends on the "gravitational anomaly" $k\pmod
3$. In particular, the conjecture that holomorphic permutation orbifolds are
non-anomalous and therefore a stronger conjecture of M\"uger about braided
crossed $S_n$-categories arising from permutation orbifolds of completely
rational conformal nets are wrong. More general, we show that cyclic
permutations of order $n$ are non-anomalous if and only if $3\nmid n$ or
$24|c$. We also show that all cyclic permutation gaugings of
$\mathrm{Rep}(\mathcal A)$ arise from conformal nets.
