Homomorphisms and rigid isomorphisms of twisted group doubles

We prove several results concerning quasi-bialgebra morphisms ${\mathcal{D}}^{\omega }(G)\to {\mathcal{D}}^{\eta }(H)$ of twisted group doubles. We take a particular focus on the isomorphisms which are simultaneously isomorphisms $\mathcal{D}(G)\to \mathcal{D}(H)$. All such isomorphisms are shown to be morphisms of quasi-Hopf algebras, and a classification of all such isomorphisms is determined. Whenever $\omega \in Z^3(G\mathrm{/}Z(G),U(1))$ this suffices to completely describe $\mathrm{Aut}({\mathcal{D}}^{\omega }(G))$, the group of quasi-Hopf algebra isomorphisms of ${\mathcal{D}}^{\omega }(G)$, and so generalizes existing descriptions for the case where $\omega$ is trivial.