Bijections which preserve blocking sets

We consider the following question: Given a family A of sets for which A-blocking sets exist, is it true that any bijection of the set of points which preserves the family of A-blocking sets must preserve A? Using a variet of techniques, we show that the answer is 'yes' in many cases, for example, when A is the family of subspaces of fixed dimension in a projective space, lines in an affine plane, or blocks of a symmetric design, but that it is 'no' for lines of an arbitrary linear space.