A couple of things here don't seem right. A Fox Model BSE would have ejectors, not extractors. Also, I don't believe Savage began putting serial numbers on them until the Gun Control Act of 1968 required it.
Fox Model B – The Fox Model B was added to the 1940 Fox catalogue put out by Savage. The retail price was $25. The Model B was offered in 12-, 16-, and 20-gauges and .410-bore, and had a color case-hardened frame. By January 22, 1945, the retail price of the Model B was up to $34.50. After WW-II the Springfield 311 (1947) and then the Stevens 311 (1948 and on) were built on a version of the Model B action. The pre-WW-II Springfield No. 311, as well as the Riverside and Springfield No. 315, the Stevens No. 330, and numerous “trade brand” guns were all built on Steven’s G.S. Lewis’ Apr 20, 1915, patented action.
Through the 1954 catalogue the Model B remained a double trigger, plain extractor, slim forearm utility gun. In the 1955 catalogue Savage added a Model BST which got a non-selective single trigger and a beavertail forearm. The offerings were the same for 1956 thru 1958.
By the 1960 catalogue the B and the BST got ventilated ribs. There were no changes in the 1961 catalogue.
For 1962 a B De Luxe in 12-gauge was added with a satin chrome receiver, white-line grip cap and butt plate, checkered side panels, beavertail, ventilated rib, and gold plated non-selective single trigger. They also gold plated the trigger on the BST. There were no changes for 1963.
For 1964 the De Luxe became the BDE, and included automatic ejectors, and also was available in 20-gauge chambered for 3-inch shells. The B and BST remained unchanged.
For 1965 the B and BST 20-gauges got 3-inch chambers.
For 1966 the satin chrome BDE was gone. There was a BSE with ejectors, non-selective single trigger, satin black finish, white-line butt plate and grip cap, beavertail and ventilated rib. The Model B had double triggers, was case hardened, had a ventilated rib, and now a beavertail. Both guns now featured impressed stamped checkering. For 1967, 8, 9 and 1970 the offerings were the same. The gun control act of 1968 required Savage to start putting serial numbers on these guns.
By 1971 there was a gun called the Savage 550 which was just like the BSE but had a case hardened receiver, and was offered in 12- and 20-gauges. The text said it had precisely engineered barrels resulting in perfect balance. In the picture it appears to be a mono-bloc design?!? Also grip caps are gone from the B and BSE. Offerings for 1972 were the same – the 550 listed at $174.95, the B-SE listed at $164.95, and the B listed at $139.95.
By 1973, the BSE and 16-gauges are gone, but there is a 24-inch barrel B in 12-gauge and 20-gauge, both with 3-inch chambers.
For 1974 the Savage 550 is gone and the B-SE is back. The B-SE 12-gauge has 2 ¾ inch chambers and the 12-gauge B and 24” Fox B have 3-inch chambers. All the 20-gauge and .410-bore are 3-inch chambered. No changes for 1975, 6, 7, or 8. For 1979 the grip caps are back and the double triggered B is offered with ejectors as the BE.
For 1980, cut checkering is back, but the 24-inch barrel offerings are gone – B, BE and B-SE. In 1981 the BE was gone, but the FA-1 and the FP-1 Fox autoloader and pump appeared. For 1982 the Fox offerings were B, B-SE, FA-1 and FP-1.
For 1983 the only Fox offering was the B-SE. The B-SE remained through the 1987 catalogue when its list price was $525. By the 1988 catalogue the Fox name was gone.