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

Homomorphisms and rigid isomorphisms of twisted group doubles