The 22 Hornet is on the ragged edge bottom for coyotes. It's about ideal for cats, fox, coons, beaver, badger, and similarly sized creatures. 22 Mag is even further down on the ragged and tattered edge for coyotes, and will be even nicer to the other game.

There's a painful reality with which you'll have to come to grips, however - groundhogs are d@mned small animals. I'm not sure why in the world anyone would put in the energy to tan one. Even a 22mag can make a mess of such small rodents, so the only real option there, if you want pristine hides, is 22LR or 17HM2, which will put you firmly too small for coyotes, and only suited for anything else on your list at the shortest of ranges. You're talking animals which span from 5-6lbs average up to 40lb+ animals, and based on how these are hunted, you could be talking about 20yrd shooting out to 200yrds or more... Anything which doesn't destroy a groundhog is NOT going to be effective at killing a coyote, and marginal for the rest of your list. Anything highly efficient at killing badger, coon, & beaver sized game will be too much for the groundhogs and too light for the coyotes. Anything really effective at killing coyotes will be over powered for the rest of your list, and will blow groundhogs to bits with anything except for ball ammo (not legal for use in all states).

So personally, I'd pick up a 22LR for the groundhogs and a 22Hornet for the rest, and you'll be quite well suited.

17 Hornet is an option, but is less effective on coyotes, and the higher velocity, lighter constructed 17cal bullets are prone to do worse damage on smaller game like groundhogs.