  In this paper we present a graph theoretic construction of Steiner quadruple
systems (SQS) admitting abelian groups as point-regular automorphism groups.
The resulting SQS has an extra property which we call A-reversibility, where A
is the underlying abelian group. In particular, when A is a 2-group of exponent
at most 4, it is shown that an A-reversible SQS always exists. When the Sylow
2-subgroup of A is cyclic, we give a necessary and sufficient condition for the
existence of an A-reversible SQS, which is a generalization of a necessary and
sufficient condition for the existence of a dihedral SQS by Piotrowski (1985).
This enables one to construct A-reversible SQS for any abelian group A of order
v such that for every prime divisor p of v there exists a dihedral SQS(2p).
