Coordinate transitivity of a class of extended perfect codes and their SQS

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 isomorphism classes of these SQS's 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 $\mathrm{GL}$-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.