The category $\mathcal{C}_l$ of tilting modules of the quantum group $U_q(sl_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when the quantum parameter $q$ is a root of unity. The module categories of $\mathcal{C}_l$ (or equivalently, quantum subgroups of $U_q(sl_2)$) are classified by ADE diagrams. See for example, section 6 of Ostrik.

Let $K_0(\mathcal{C_l})$ be the Grothendieck ring of $\mathcal{C}_l$. The action of $\mathcal{C}_l$ on a module category by tensor product on the objects defines a based module $M$ over $K_0(l)$. The generating element of $K_0(\mathcal C_l)$ acts as a matrix $A$ on $M$, where $A$ is the adjacency matrix of the corresponding ADE Dynkin diagram.

The ADE Dynkin diagrams have graph symmetries: the A-type and E_6 graphs can have the order of the vertices reversed, the D-type graphs can switch the two short legs, and the D_4 graph exhibits a full $S(3)$ symmetry. All of these symmetries induce based module symmetries on the corresponding based module $M$.

In what ways can the symmetries on the based modules be extended to symmetries of the corresponding module categories? Can you construct module functors from the module category to itself that induce this symmetry on the based modules? If not, is there some lesser structure than a module category but greater than a based module that can be preserved under these symmetries?