Here is what IGC has on them:
F P Baker & Co Ltd
Address1 1 Golden Square
Address2 2 Upper James Street, Golden Square
Address3 13 Panton Street
Address4 Warwick Street
City/Town London
Country United Kingdom
Trade Gunmaker
Dates 1865-1970?
Notes
F P Baker established his business in 1865 at 1 Golden Square, London, but the building appears to have run through to Warwick Street. The firm was a department store with a sporting goods and a gun department; they bought most of their guns in Birmingham, Webley, Westley Richards and Skimmin & Wood were known suppliers. At the turn of the century the firm were located at 2 Upper James Street, Golden Square, Regent Street. In about 1907 the gun department was located at 13 Panton Street. At some time, probably in the 1920s, Leonard Vidal retired from the Indian Army and bought the firm. Montagu Richard Reynolds Vidal was a director of the firm for many years, probably up to 1962 when he died.
In 1965 the firm published a centenary brochure entitled "1865-1965, A Century of Personal Service, Bakers of Warwick Street" (Warwick Street lies parallel with and between Regent Street and Golden Square).
Internet Gun Club has some details of serial numbers which we have not published. Please send details of your gun and its serial number by email to archives@internetgunclub.com and we will reply with what information we have.
Tim