Either the .243 Winchester or the 6mm Remington would do the job. Both are extremely accurate.
The .243 Win. is a .308 case necked down to .24 caliber. The 6mm Remington (aka .244 Remington) is a 7mm Mauser case necked down to .24 caliber. The 6mm Rem. can be pushed to somewhat higher muzzle velocities and is, therefore, a bit flatter shooting than the .243, but not enough to make much difference.
The .243 Win. is, by far, more popular... not because it's better, but because Big Green ("Remington") screwed up and originally made their rifles for the .244 Remington rifle barrels with a 1:12 turns which would NOT stabilize 100 grain bullets (used for deer) whereas the .243 Winchester rifles WOULD stabilize them due to their 1:10 turn barrels.
Remington corrected their mistake by re-designating (NOT "re-designing") their ".244 Remington" as the "6mm Remington" (exactly the same cartridge in all ways) and putting 1:9 twist barrels on their rifles for the 6mm Remington caliber, but it was TOO LATE.
The .243 Winchester got a big "jump" on the 6mm Remington... and it appears the 6mm Rem. will never "catch" the .243 Winchester in popularity even tho' it's a slightly superior round.
However, neither the deer nor the groundhog will know the "difference" between the .243 Win. and the 6mm Rem., so you can take your pick and it won't be wrong... EITHER ONE is an excellent choice.
Strength & Honor...
Ron T.