  For a finite group $D$, we study categorical factorisation homology on
oriented surfaces equipped with principal $D$-bundles, which `integrates' a
(linear) balanced braided category $\mathcal{A}$ with $D$-action over those
surfaces. For surfaces with at least one boundary component, we identify the
value of factorisation homology with the category of modules over an explicit
algebra in $\mathcal{A}$, extending the work of Ben-Zvi, Brochier and Jordan to
surfaces with $D$-bundles. Furthermore, we show that the value of factorisation
homology on annuli, boundary conditions, and point defects can be described in
terms of equivariant representation theory. Our main example comes from an
action of Dynkin diagram automorphisms on representation categories of quantum
groups. We show that in this case factorisation homology gives rise to a
quantisation of the moduli space of flat twisted bundles.
