I'll look again tomorrow after I get unthrottled internet service, right now I can only look on my phone. From what I can see, the stock doesn't look bad enough to work on if you plan to use it for hunting.
If I worked on it, I would plug the hole in the forearm with a piece of walnut. I would mask the checkering, sand it to 320 grit, maybe 400 grit, along with whiskering, and then I would finish it with either Gun Sav'r or Minwax aerosol polyurethane applied in a thin application and rubbed in with a cotton rag. Three applications will be enough, and you're done at the end of the day.
I'm done screwing around with a zillion applications of rubbed out oil finishes on a gun I intend to use, and everyone will admire your hand rubbed satin polyurethane oil finish, just don't tell them what you used or that their newer guns with hand rubbed oil finishes are finished the same way.
All I can see at the butt plate is a dark looking region that is typical for a spot with wood filler, I'll get a better view tomorrow. A careful repair can be completely invisible, the forearm is easy, the butt might not be easy.