The logo looks to be "CI" with the "C" resembling a hunting horn transfixed by a gnarly mountainside tree and the "I" dotted by a flying duck. (It possibly could be "CF," "OI" or "OF.") It does not look French to me. Looks more Central European. Also, it's pretty sophisticated work and looks modern. There is writing below the symbol - can't make out the script. Might it be Cyrillic?
Hunting horns do appear on some French guns - here is the butt plate of my 1906 Gerest Berthon with a horn motif - but nothing as complex as that.