From my research it appears that in 1883 William Schaefer's elder son, John Schaefer, was brought on as a partner in the business. Interestingly, it appears that the younger son, William, was never made a partner. I have only seen the "Schaefer & Son" name in period advertisements and business records, never "Schaefer & Sons". Similarly, I have only seen the "& Son" markings on the guns themselves.

The hammerless Schaefer & Son guns appear to be sourced from two makers - Westley Richards in England and Lindner in Suhl. The Westley sourced Schaefer actions have a flat action while the Lindner sourced guns are scalloped (along with other stylistic differences). From an admittedly small sample, I would also make the observation that the Westley Richards sourced guns usually feature engraving while the Lindner guns often have no engraving

Ken