I think one of the reasons for confusion is that before update 1.4, there was a different method for acquiring a Sol permit, and a lot of older posts on this subject are still flying around the internet.
Anyway a summary of my video below(which explains in more detail) is that you just simply do Federation missions to improve your Fed rank until it reaches Petty Officer. Then you will automatically get a Sol permit.
To increase Fed rank you will need to do a Federation Navy mission, but these only appear once you are close to going up a rank. And you do this by improving your reputation with minor faction allied with the Federation.
That's pretty much it. Keep doing the missions and you will eventually get your permit,