While the 1903 was initially a product of the Springfield Armoury, which started design work in 1900, it was not until many variations later that Remington Arms was comissioned with remanufacturing the 1903, and they released the M1903A3 in 1942. The M1903A4 was released shortly thereafter, having it's iron sights replaced with M73B1 2.5X (4.76X in the game.) telescopic optical sights, and afterwart the M1903A4 remained in service until the late 1960s.
M1903A3 Springfield on Wikipedia