  Bujalance, Costa and Izquierdo have recently proved that all those Riemann
surfaces of genus $g \ge 2$ different from $3, 6, 12, 15$ and 30, with exactly
$4g$ automorphisms form an equisymmetric one-dimensional family, denoted by
$\mathcal{F}_g.$ In this paper, for every prime number $q \ge 5,$ we explore
further properties of each Riemann surface $S$ in $\mathcal{F}_q$ as well as of
its Jacobian variety $JS.$
