That was fun for me in the 70's. .222 Rem in a bolt gun was considered "THE" way to go then. Anything flat shooting and accurate is fine. A 40-50 grain bullet is plenty, you are not going to need a .220 swift.
If I was looking for an excuse for a new Varmint rifle, I would look pretty hard at that .204.
How far away do you want to shoot them? .17 HMR, .22 WMR are quite adequate out to 100yrs or so. If you can get inside 75yrs or so .17 mach2 or .22LR HVHP works fine. .222 or .223 for farther out. So many good choices in the .20 and .17 cal Wildcats. Then there are the .22 hornet, .218 Bee, the list is voluminous.
I've used the .22LR to great effect. A little known fact, however, is that the 223 was actually invented as a GH caliber. True story. Basically what happened is that one of our FOBs in Korea was overrun by the critters. Sure, there are stories of rabies and even a zombie virus, but truthfully, no one really knows what happened. I digress. At any rate, DoD went after the little furry savages to the point of commissioning Remington to create a purpose built caliber. The AR was then invented to take advantage of the new round originally dubbed the "kritter killer." Early use in Southeast Asia showed increasing possibility for the AR so a new, more powerful round was developed. We now know this cartridge as the NATO 5.56x45.
Coming full circle, the popular AR is now nicknamed the "zombie" gun and is chambered in 223 Wylde. Most people don't realize that the zombies referenced are the GHs from a lonely hill and a desperate battle in Korea.
Thats is why if I ever need to take arms agsin to face the tunneling son of a beaver, I'm going out with an AR. It's about justice, poetic justice.
Anything that makes a clean kill is ok with me. I've killed them with 22, 38, 357, 44 in handguns. 22, 223, 243, 308, 45/70 in rifles. 12 and 20 gauge in shotguns. You just can't beat shooting a groundhog, regardless of what you use. Part of the fun is deciding what you want to use, and then getting close enough to make a clean kill with it.
depends on your location. I have been very successful with my 17 HMR when shooting closer to town. Works great on calm days and it is a very flat shooter.
If you want a good challenge use a bow, it is fun. After the first couple the challenge will get harder. They get smart. I tried a pellet gun but the one I used was not strong enough. When you hit one they squeeled like a pig. I had really good success with 55 grain soft points from a .223. I had good luck with 00 buck on some too. Good for when you get some that run from hole to hole. It can be fun using big guns that you wouldn't normally use. Dead is dead so who cares if you decide to use something most would call over kill.
I use .243 on my place. I don't really have to judge distance much with it. Turns them to mush. I've got an old 788 Remington in .222 that does well out to about 250yds max, but when they're way across the field the .243 can reach out and give 'em a kiss.
Dad used an old sporterized Eddystone .30-06 with Remington Accelerators. I watched him center punch ground hogs on my uncles farm at 250+ on more than one occasion. I prefer a .223 with a good ballistic tip or Hornady VMAX.