No, it is not Indio National Forest. It is Inyo National Forest on the east side.
You do not have to get a Mount Whitney Trail permit (entry), which is hard to get, but you still have to have a Mount Whitney Trail permit (exit), which is easier to get, but you still have to have it.
Everybody and his dog will be hiking southbound on the JMT, wanting to go up Mount Whitney, and then they will have to exit on the Mount Whitney Trail. So, don't expect this to be a wilderness experience. Well, actually no dogs, although I have a photo of a service dog on the Mount Whitney Switchbacks one time.