This is the usual repair and woodworker's supply even make a taper tool and provides the plugs. The plugs are tapered like a pencil point. I have only seen this problem on older guns and seems to be mostly what they call dry rot in wood. However, I have seen some Remingtons that have a great stock finish but raw wood under the butt plate. I sometimes coat these butt ends with epoxy or a good spar varnish. Take care to mark and drill the new holes as your butt plate won't be veryy forgiving. I know some folks to put beeswax on the threads of the mounting screws and other put epoxy, I guess in part to seal this area as well. I use beeswax. Understand that I am not a gun smith but a tinkerer. I do build M/L,s and rework stocks. Good luck..
Be Safe !!!