I've shot a few foxes in my time and have come to the conclusion, along with several other friends of mine who call foxes, that there is "NO" .224 diameter bullet (22 caliber) that is fur friendly at centerfire speeds. I have destroyed fur with a 22 Hornet and 35 gr V-max, Gray Fox take it tough with even a 22 Magnum and 33 gr V-Max at called ranges. When hunting Reds you will need the reach of a centerfire, they don't normally come as close when called, so I suggest using a 17 caliber, either the new 17 Fireball, or the 17 Remington, I have been using a 17 Ackley Hornet with alot of success this year so far.
Your odds are better at killing a coyote with a well placed 17 Fireball/17 Remington shot than they are of not harming the fur on a fox with a 22 caliber centerfire.
This guy was shot with a 17 Ackley Hornet 20 gr V-Max @3850fps, at approx 125-150 yards, can't even find the entrance. The blood on his head is from another fox that was laying on top of him in the bed of my truck. These thoughts are just mine, others may vary, and they are entitled to their opinions too.