Even if it is not a clean break, as long as the wood is clean, a good polyurethane glue will repair that if clamped properly.
Personally, I would remove it from the rifle, glue it back with polyurethane glue, and if I didn't want to refinish, then I would be super careful about the edges. It foams and expands.
I'd probably refinish it and any extra pieces missing from the break, I would fill with a black or heavy contrasting epoxy to make the missing pieces pop. It's the same thing we do with mesquite or tables where wood is checked. Fill it. Accentuate it. The worst repairs to me are the ones where someone tried to make it disappear and failed. Make it clean and solid and a good battle scar. It's not unlike Japanese Kintsugi.
The other option will also require refinishing, but you can cut donor splinters of wood from inside the stock. Those can be glued into the missing areas and sanded smooth. You'll never see it since the wood is the same and it will take a stain the same as everywhere else.