There were no "Model Numbers" on Remington Arms Co. doubles. The common terms in use today, Model 1889, Model 1894 and Model 1900 were only used by Remington Arms Co. on parts lists. The Remington Hammerless Double Barrel Shotgun (aka Model 1894) was introduced in their October 1894 catalogue, and came with increasingly finer Damascus barrels as the grade went up from A- to EE-Grade. During 1897 Remington began to offer steel barrels on their hammerless double. They offered two qualities of steel. Remington Steel barrels were offered on the entry level A-grade guns at the same price as the regular two-stripe Damascus barrels. Ordnance Steel was offered on the A- and B-Grade guns at a $10 premium over their regular Two- and three-stripe Damascus barrels and at the same price as the Damascus barrels on the C-, D- and E-Grades. What the difference is, if any, between the Remington Steel on their hammerless doubles and the decarbonized steel on their hammer double has long been an unanswered question.
Circa 1900, Remington Arms Co. introduced a cheaper version of their hammerless double called the K-Grade (aka Model 1900). Originally they introduced the K-Grade with Remington Steel barrels and the KED-Grade with ejectors and two-stripe Damascus barrels. A few years later they added a plain extractor Damascus barrel version called the KD-Grade, and finally they added the KE-Grade with Remington Steel barrels and ejectors.
For 1906, Remington Arms Co. inserted a lower priced "Trap" Gun into their line up between the A- and the B-quality guns and called it the Grade "F.E." Trap Gun --
The Grade "F.E." Trap Gun is the only gun I've seen marked with F E G on the underside of the barrels. Mine that dates to 1906 is marked /// F E G 13. I know of another one that dates to the 1910 sell-off of break-action guns that is marked /// F E D1.