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 F_gFg in the moduli space M_gMg: 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 F_gFg. The set of real Riemann surfaces in F_gFg consists of three intervals its closure in the Deligne–Mumford compactification of M_gMg is a closed Jordan curve. We describe the nodal surfaces that are limits of real Riemann surfaces in F_g