Question: Meaning of SymmetryGroup in the Logic package

I am trying to understand the SymmetryGroup returned in the Logic Package. The help page says "The group is a permutation group; its elements are those permutations which preserve the Boolean structure of expr." [my bold], but later the definition is given as "A symmetry of a Boolean expression expr is a mapping f of each variable to some other variable or negated variable, such that the image of expr after applying f to each of its variables is a Boolean formula which is equivalent to expr." Is logically equivalent meant here, or something else? The help page examples don't answer this question.

The following example shows that a group permutation does not lead to a logically equivalent statement as I was expecting - is this a bug, or am I expecting too much here?

Download LogicTest.mw