Sorry if I annoy anyone with this long post. I personally must disagree with conan on this, the 5.56 (.223) is very unstable and does not work well at ranges +300metres
It's used in the m16/ar15 because of it's instability. The bullet is smaller lighter and "bounces around" inside a living target.
Besides, unless you keep an ar15 completely clean, they misbehave.
Not that a cute little ground hog is a dangerous enemy, but having trained extensively with them, I don't like 'em much.
Anyway
back to the topic at hand I'd say .243 would be my choice. A little bigger and more accurate. Heavier so less wind problems but still light enough not to worry too much about bullet dropping. Never tried 22-250, so I don't know much about it.
Assuming maybe .243 is too expensive (C'mon guys, it a GROUND HOG, the fate of the world doesn't hang in the balance if you miss) would be 7.62NATO (.308) Cheap, and available everywhere in the world, Hoorah.
Bullet is nice and heavy (Not TOO heavy) so it remains stable and accuate 800metres (Rifle dependant) Though, not being the greatest shot in the world, I could probably only hit a ground hog at 400m or so with one, because the bullet drops alot compared to a small sized cartridge.
Don't forget a nice high powered scope so you can see the cute bastards terrified looks as you turn them to "ground sausage"