  In this paper we study marked numerical Godeaux surfaces with special
bicanonical fibers. Based on our construction method of marked Godeaux surfaces
we give a complete characterization for the existence of hyperelliptic
bicanonical fibers and torsion fibers. Moreover, we describe how the families
of Reid and Miyaoka with torsion $\mathbb{Z}/3\mathbb{Z}$ and
$\mathbb{Z}/5\mathbb{Z}$ arise in our homological setting.
