Higher orbitals of quizzy quantum group actions

The hyperoctahedral group $H_N$ is known to have two natural liberations: the "good" one $H_N^+$, which is the quantum symmetry group of $N$ segments, and the "bad" one $\bar{O}_N$, which is the quantum symmetry group of the $N$-hypercube. We study here this phenomenon, in the general "quizzy" framework, which covers the various liberations and twists of $H_N,O_N$. Our results include: (1) an interpretation of the embedding $\bar{O}_N\subset S_{2^N}^+$, as corresponding to the antisymmetric representation of $O_N$, (2) a study of the liberations of $H_N$, notably with the result $<H_N^+,\bar{O}_N>=O_N^+$, and (3) a comparison of the $k$-orbitals for the inclusions $H_N\subset H_N^+$ and $H_N\subset\bar{O}_N$, for $k\in\mathbb N$ small.