LAUSR

We prove several results concerning quasi-bialgebra morphisms D ω ( G ) → D η ( H ) ${\mathcal {D}^{\omega }(G)\to \mathcal {D}^{\eta }(H)}$ of twisted group doubles. We take a particular focus on the isomorphisms which are simultaneously isomorphisms D ( G ) → D ( H ) ${\mathcal {D}(G)\to \mathcal {D}(H)}$ and completely determine them. Whenever ω ∈ Z 3 ( G / Z ( G ), U (1)) this suffices to completely describe Aut ( D ω ( G ) ) ${\text {Aut}(\mathcal {D}^{\omega }(G))}$ , the group of quasi-Hopf algebra isomorphisms of D ω ( G ) ${\mathcal {D}^{\omega }(G)}$ , and so generalizes existing descriptions for the case where ω is trivial.