G.N.
Some research I did on the topic:
How do you remove permanent marker stains from carpet? Well, the bad news is that it's called "permanent" marker for a reason. Depending on how new and how deep the stains are, you may not be able to remove them completely.
You can almost always lighten stains, though, and sometimes make them unnoticible. There are other things you can do to hide the stain as well. Start with the method here to get out as much as you can.
First, thoroughly vacuum the area. This removes any loose dirt, dust or other substances that might give you trouble when you work on the stain.
For the carpet stain removal solvent, use rubbing alcohol or a non-oily hairspray. Use a small amount on a clean white cloth, and dab at the stain. You should see transfer of the stain to the cloth. Use another dry cloth to blot up the solution between applications.
Just apply the solution and blot it up several times. Then rinse the spot with clean water and blot that up. Finally, dry the area thoroughly.
Get Out The Remaining Stain
If you couldn't get it all out, and if the stain isn't too deep, it's time to cut out some carpet - just a little. Try snipping away the stained edges with small scissors or with a razor blade or exacto knife. Use tweezers to carefully pull out severely stained fibers.
When you try the latter, you might want to remove clean carpet strands from another, hidden area in the room, and then glue them into the stained area. Then leave a heavy object on the spot for several days before walking on it. That's how you remove permanent marker in all but the worst cases.