Prokhorov and Shramov proved that the BAB conjecture (which Birkar later proved) implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of semi-simple groups (i.e. those with no non-trivial normal abelian subgroups) acting faithfully on n-dimensional complex Fano varieties, and this bound only depends on n. In this talk, we investigate the consequences of an action by a particular semi-simple group: the symmetric group. This work is joint with Louis Esser and Joaquín Moraga.