  In 1988, Duval introduced the concept of directed strongly regular graphs,
which can be viewed as a directed graph version of strongly regular graphs.
Such directed graphs have similar structural and algebraic properties to
strongly regular graphs. In the past three decades, it was found that Cayley
graphs, especially those over dihedral groups, play a key role in the
construction of directed strongly regular graphs. In this paper, we focus on
the characterization of directed strongly regular Cayley graphs over more
general groups. Let $G$ be a non-abelian group with an abelian subgroup of
index $2$. We give some necessary conditions for a Cayley graph over $G$ to be
directed strongly regular, and characterize the directed strongly regular
Cayley graphs over $G$ satisfying specified conditions. This extends some
previous results of He and Zhang (2019).
