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