I just bought a 4th Gen burnside carbine and the stock needs some repair. Is there anyone here who can help me? I would like to keep the original stock as it has the same serial number as the rest of the gun.
Personally I'd leave it, it's an old warhorse and it will have scars. If you must repair it, you'll have to clean the existing wood, glue on a piece of walnut, shape to correct profile, then try to age or blend the repair so it doesn't stick out. It can be done, but takes practice.
This needs to be done professionally. Burnsides are a collector's item, depending on the model.
Having said that, stock repair is possible. I have just done my first stock repair on an old Gras rifle. It was an old rifle - probably colonial in the end - with a repaired stock, water- and oil-damaged to the point of carbonisation! It was cheap and the metal wasn't much better, very corroded and pitted, so it was worth a try. It was unusual in that it was all same-serial so a new stock was out of the question as the butt is serialled too. The two shoulders at the back of the action were charcoal. After soaking the stock and drying off, I noticed water leaking out of the stock in front of the trigger guard - the wood around the cleaning rod hole had rotted and was letting out any trapped water. Since it was not really a rarity - I got stuck in. BTW - it ain't no picnic!
Lodgewood does excellent work, as stated above, but can be pricey. I'm not sure where you're located, but another good alternative is Clay Smith in Williamsburg, VA. He's a former gunsmith for Colonial Williamsburg and specializes in antique weapons. He recently replaced the forend on a Trapdoor Springfield for me, and the job is so good you can't tell where the old wood ends and the new wood starts. His website is https://claysmithguns.com/.