We determine, for all genus g≥2g≥2 the Riemann surfaces of genus g with exactly 4g automorphisms. For g ≠ 3,6,12,153,6,12,15 or 30, these surfaces form a real Riemann surface FgFg in the moduli space MgMg: the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the family. Furthermore we determine the topological types of the real forms of real Riemann surfaces in FgFg. The set of real Riemann surfaces in FgFg consists of three intervals its closure in the Deligne–Mumford compactification of MgMg is a closed Jordan curve. We describe the nodal surfaces that are limits of real Riemann surfaces in Fg