From Ponce. (1968) (public domain):

Unexpanded basidiocarp globose, at first submerged, medium or large
size; exoperidium split to about the middle into 4-5 rays; the outer
layer remaining as a hollow cup in the substratum; the inner fibrous
and fleshy layers becoming strongly fornicate, attached at tips to the
basal cup, rays firm, thick, brown; fleshy layer dark brown and
adherent or partly peeled off, in old plants worn away; base strongly
convex; endoperidium subglobose or urnshaped to depressed, dark brown,
finely velvety on surface, constricted near the base so as to form a
ring-like apophysis above the pedicel; pedicel about 2-3 mm. long;
mouth naked, conical or mammiform, tubular, apex fibrillose or
lacerate; gleba ferruginous; columella long-elliptical, 2-3 mm. high;
spores globose, 5-6um in diameter, umber, verrucose, reticulate;
capillitium threads 10-12um thick, about as dark as the spores; walls
roughened, tapering, with ramifications at the ends.

Europe, North America, South America, Australia, North Africa, South
Africa, and Hawaii.

Solitary on vegetable debris on the ground.

This species is characterized by the fornicate exoperidium and has
been confused with G. minimum in Europe and North America, and with
G. radicans in North America. It may be distinguished from them by its
naked mouth. This species has been called G. fornicatus (Huds.) Fr., a
name which is untenable. The species is very similar to G. rufescens
in fundamental characters, such as mouth, surface at inner peridium,
separating layers of outer peridium, and concave base when expanded.

Geastrum quadrifidum Pers. Syn. Fun. 133. 1801.
G. quadrifidum f. fenestratum Pers. Syn. Met. Fung. 133. 1801.
G. quadrifidum x majus Abb. & Schw. Consp. 79. 1805.
G. quadrifidum DC. Fl. Fr. 2:267. 1815.
Geaster fornicatus Fr. Syst. Myc. 3: 12. 1829.
G. macowani Kalchbr. Fungi Macowaniani, Grev. 10: 108. 1881.
G. marchius P. Hemm. Nat. Pflanz. 1: 231. 1901.
G. fenestratus Lloyd, Myc. Writ. 1: 70. 1901.
G. fenestriatus (Pers.) Cunn. Proc. Linn. Soc. NSW. 51: 89. 1926.
Plecostoma fornicatum Desv. Journ. Bot. Paris 2: 101. 1809.