The road to the South Grove can be closed due to weather or debris on the road. Call the park ahead of your drive to be sure.
If this is an add-on to the South Grove Loop, stay straight on Forest Rd 5N05X at the finishing intersection instead of turning left back to the bridge and your car. If you're doing just this loop, come up from the bridge and make a hard left at the intersection.
Continue along 5N05X, make sure to watch for barbed wire cattle gates. As a general rule, close them behind you if they were closed when you came up on them & leave open gates open behind you. Keep going on this trail until you hit the OK Corral. Shortly after the Corral stay left at the next split and ride down to a wooden bridge crossing to the north side of Beaver Creek.
Head back staying on 5N78. More cattle gates, and the road will climb a bit before coming back down to a 'T' intersection. Head to the right and you'll be back at your car right away.
I can recommend the National Geo/ Trails Illustrated Map #808 which will help you with this and any other fire roads in the area you wish to explore.