It is well-known that the symmetric group S4 has two Schur covering groups, S4-tilde and S4-hat. There are explicit presentations for both groups, and we know that S4-hat is isomorphic to GL(2,3). Question: Is there an alternative, explicit description/realization of S4-tilde?

The lowest degree d for which S4-tilde can be realized as a transitive permutation group is d=16. Is there a concrete (geometric?) construction of this permutation action? (other than via coset action...)

According to wiki, S4-tilde can be embedded inside GL(2,9). What is a good way to "see" this (geometric?) action? (transitive, I presume?)

Last but not least: Is there a "geometric" way to think about/realize the two Schur covers of S_n for general n? And what the six-fold covers of A_n for n=6,7?

Thanks!