Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a spherical fusion category to a ribbon fusion category. Interesting means, that they should differ from those functors obtained from group homomorphisms in the following way:
Let $\phi: H \hookrightarrow G$ be an inclusion of finite groups. Define the functor $\phi^*: \rho \mapsto \rho \circ \phi$ on the representation categories. Denote the regular representation by $\Omega_G$ or $\Omega_H$. Then $\Omega_H$ is a subobject of $\phi^* \Omega_G$, more specifically, $\phi^* \Omega_G \cong \oplus^n \Omega_H$, with $n = \lvert G/H \rvert$ the number of cosets.
Are there pivotal functors $F: \mathcal{C} \to \mathcal{D}$ where $\mathcal{C}$ is spherical fusion and $\mathcal{D}$ is ribbon fusion, such that $\Omega_\mathcal{D}$ is a subobject of $F\Omega_\mathcal{C}$, but $F\Omega_\mathcal{C}$ is not isomorphic to $\oplus^n \Omega_\mathcal{D}$?
$\Omega_\mathcal{C}$ denotes the sum over all the simple objects in $\mathcal{C}$.
(This is related to my earlier question here: Is the category of spherical fusion categories regular? (i.e. is image factorisation possible?))
