A different kind of project...help needed

[Flying10uk]

Looking at standard die sizes though maybe those outermost threads are 28 per inch.

If you look at your first pic on the bottom row of pics, post 1, you can see that, on the outer sleeve, at the end of the threaded portion, there is an "undercut" suggesting that the thread was turned on using the lathe's "lead screw", as would be expected. This is for the "run-out" of the cutting tool so that when the cut is finished the tool is not actually cutting any metal. If this "undercut" was not there normally the very tip of thread cutting tool gets broken off.

Manual lathes capable of screw cutting normally come either with an imperial lead screw, on an imperial lathe, or a metric lead screw, on a metric lathe.

With a turned on thread you can have almost any threads per inch/pitch you want subject to the capabilities of the lathe. Some lathes have greater capabilities than others.

Cutting internal threads, using a lathe, is more difficult than external threads, partly because you can''t see what is going on as the tool disappears, under power, into the workpiece.

Some people, such as a machine shop, may decide to produce the parts and cut the threads using a CNC lathe.