I thought at first that you meant F.T.Baker who was a Birmingham maker and generally supplied quality guns. I looked in Nigel Brown's British Gunmakers and this is the bit on F.P.Baker.

F.P.Baker, 13, Panton Street, London SW1. Firearms bearing this name and address proved after 1907 have been seen but no other ref. found.

That label you have tells more than the book!

I would say that the gun was actually made by Webley & Scott and retailed by Baker. Lagopus.....