Metallic Mn-based alloys with a nearest-neighbor Mn-Mn distance greater than 0.4 nm exhibit large, well-localized magnetic moments. Here we investigate the magnetism of tetragonal ${\mathrm{Au}}_{4}\mathrm{Mn}$ with a Curie temperature of 385 K, where manganese has a spin moment of $4.1\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}$ and its orbital moment is quenched. Since 80% of the atoms are gold, the spin-orbit interaction is strong and ${\mathrm{Au}}_{4}\mathrm{Mn}$ exhibits uniaxial magnetocrystalline anisotropy with surface maze domains at room temperature. The magnetic hardness parameter of 1.0 is sufficient to maintain the magnetization along the $c$ axis for a sample of any shape. Au also reduces the spin moment of Mn through $5d\ensuremath{-}3d$ orbital hybridization. An induced moment of $0.05\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}$ was found on Au under a pulsed field of 40 T. Density functional theory calculations indicate that the Mn-Mn exchange is mediated by spin-polarized gold $5d$ and $6p$ electrons. The distance dependence shows that it is ferromagnetic or zero for the first ten shells of Mn neighbors out to 1.041 nm (64 atoms), and very weak and oscillatory thereafter.