 Since "Q" is not split over "k", an auxiliary argument with special orthogonal groups over "k" forces "K" to be a field (rather than a product of two copies of "k").The natural S6-action on everything in sight defines a map from S6 to the "k"-automorphism group of "X", which is the semi-direct product "G" of PGL2("K") = PGL2(9) against the Galois involution.This map carries the simple group A6 nontrivially into (hence onto) the subgroup PSL2(9) of index 4 in the semi-direct product "G", so S6 is thereby identified as an index-2 subgroup of "G" (namely, the subgroup of "G" generated by PSL2(9) and the Galois involution).