I want to compute the automorphism group of a matroid. This reduces to the following problem:

Suppose I have a list of sets {{b_11,...,b_1k},...,{b_k1,...,b_kk}} where each b_ij is in {1,...,n}. The symmetric group S_n acts on this set by: s*{{b_11,...,b_1k},...,{b_k1,...,b_kk}} = {{sb_11,...,sb_1k},...,{sb_k1,...,sb_kk}}.

I want to find those elements of the symmetric group on n elements that stabilize the entire set {{b_11,...,b_1k},...,{b_k1,...,b_kk}}.

The command GroupSetwiseStabilizer seems relevant but doesn't quite do what I want.