Coordinate transitivity of extended perfect codes and their SQS

We continue the study a class of binary extended perfect propelinear Solov'eva 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 GL-equivalence (similar to EA-type equivalence for permutations of $F^r$). As a byproduct, we are able to construct an infinite series of non-Mollard codes whose automorphism groups act transitively on the sets of codewords as well as the sets of their neighbors (1-neighbor transitive codes).