the 1903A3NM wasnt a Remington made rifle, it was built after issue, most had standard S stock. less then a handful have suvived and a real one is rare. Harrisons book has more mistakes then Poyers, though i like both books.
but , no , SC made no C stocks, most common used on the A3, is Keystone, i avoid the auctions on Fleabay that use, Type 6, or 13 ect to try and discribe what they are selling, same reason i avoid bidding on anything listed as a Sniper part that isnt, to try and get more money from someone thinking its special.