This paper is a continuation of the author's precediug note [5]. We want to study mainly about the group of equivalence classes of holomorphic Z_2-line bundles over a compact Riemann surface of genus three. To assure that an involution is holomorphic and to see explicitly an aspect of a ramification, we treat plane algebraic curves without singularity. §1 contains reformulations of some known results in convenient forms, and these are used explicitly or implicitly in §2 and Remark. Especially, a fundamental result due to A. Hurwitz is effectively used to see topological structures of surfaces. The exact sequence (3) in §2 is one of our main results. In Remark an example is given, and it is proved that there exists no holomorphic G-line bundle other than trivial bundle.