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We continue the study of the class of binary extended perfect propelinearcodes constructed in the previous paper and consider their permutationautomorphism (symmetry) groups and Steiner quadruple systems. We show that theautomorphism group of the SQS of any such code coincides with the permutationautomorphism group of the code. In particular, the SQS of these codes arecomplete invariants for the isomorphism classes of these codes. We obtain acriterion for the point transitivity of the automorphism group of SQS ofproposed codes in terms of GL-equivalence (similar to EA-type equivalence forpermutations of F^r). Based on these results we suggest a new construction forcoordinate transitive and neighbor transitive extended perfect codes.

Coordinate transitivity of extended perfect codes and their SQS