  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.
