To get back to the OP's question... there really aren't any lineages of the A.'.A.'. for the order is one and every member can trace their line back to Crowley, the head of the Order. Even according to the Book of the Law he is the sole authority on Thelema and the A.'.A.'. As such though there are several broken links in the chain.
Crowley's reference to charlatans in One Star in Sight was in reference to the then recently expelled C.F. Russell who went on to found the Great Brotherhood of God, aka the Choronzon Club, wherein he claimed to be presenting a short cut to initiation. Whether Russell was a charlatan as a magician is not the issue, Crowley was under the belief that Russell was still presenting himself as a representative of the A.'.A.'. and that was the issue.
The broken links I refer to are people who were expelled by Crowley but continued the work or people who were lost in the shuffle when their superior had resigned/been expelled etc. Another factor in this "broken chain" affair is that in a lot of cases most members of the A.'.A.'. never officially passed beyond Probationer, let alone Neophyte or Zelator. In this I am referring to those who held onto the flame of the order following Crowley's death and the disorganization that occurred in the following years under Karl Germer. He was just as lax in his running of the A.'.A.'. as he was the OTO. Even Germer's primary student never technically passed beyond Neophyte, but claimed the Zelator grade shortly before Germer's death. I think the only initiation Germer mentioned outright was Phyllis Seckler's 5=6 status and then that was a reference to being under her HGA's guidance.
So the various "lineages" exist because of the fracturing of the order under leadership that was not ready, nor willing, to bear that responsibility. It was a time where dormancy was probably best as well, cultivating a small group that would bear greater fruit in the next generation when the world might be ready. Germer I think did what was necessary with what he was given.
There are several people out there with legitimate links to the A.'.A.'. in the sense of they signed Oaths & Tasks under people who signed Oaths & Tasks that go back to Crowley but the authority of the A.'.A.'. itself as a formal organization? As a working hierarchy with a direct link to the Secret Chiefs of the order? Well, that is debateable. I have my own opinion on that issue but I tend to keep it to myself. I know who I feel is the head of the Order on the material plane based on what I have seen & read.