Mel,
I have never seen records or any other information indicating that Baker made a production 20 ga while located in Batavia. I would guess that they may have been working on it as I have seen and owned some small frame guns that are very late Batavia production. Folsom manufactured a 20 gage after they bought Baker in 1919. Not sure what you are looking at, but not all Folsom Bakers have an "F" with the serial number.
Chris