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 $\omega\in Z^3(G/Z(G),U(1))$ this suffices to completely describe $\operatorname{ Aut}(\mathcal{ D}^\omega(G))$, the group of quasi-Hopf algebra isomorphisms of $\mathcal{ D}^\omega(G)$, and so generalizes existing descriptions for the case where $\omega$ is trivial.
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