The flipping criterion for invertibility of morphisms of doubles

Let $G$ and $H$ be finite groups, and let ${\mathcal D}(G)$ and ${\mathcal D}(H)$ denote their doubles. A morphism is flippable when it satisfies the flipping condition described above, and let $\psi\in\operatorname{Hom}({\mathcal D}(G),{\mathcal D}(H))$.

Flipping criterion. The morphism $\psi$ is invertible if and only if it is flippable and sends the integral of ${\mathcal D}(G)$ to the integral of ${\mathcal D}(H)$.

This is proposed as an invertibility test in terms of flipping. The surrounding discussion notes that every automorphism is flippable, but that flippable morphisms need not be invertible, as shown by endomorphisms of ${\mathcal D}({\mathbb Z}_2)$.