Remark on Hopf images in quantum permutation groups $S_n^{+}$

Motivated by a question of A.~Skalski and P.M.~So{\l}tan about inner faithfulness of the S.~Curran’s map, we revisit the results and techniques of T.~Banica and J.~Bichon’s Crelle paper and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4,k=2$, but also to classify the automorphisms of $S_4^{+}$, describe all the embeddings $O_{-1}(2)\subset S_4^{+}$ and show that all the copies of $O_{-1}(2)$ inside $S_4^{+}$ are conjugate. We then use these results to show that the criterion we applied to answer the aforementioned question does not admit converse.