Help With Bottle Cap Thread Design

[FLSUNNOTFUN]

I'm trying to design and print parts that can screw onto the top of the bottle, just like the cap does. This is one of the cheap, thin plastic water bottles with a cap that is about half of the size of a "standard" size bottle cap. I've looked all over for thread specs on these types of caps but haven't found anything really helpful. I've gotten pretty close by trial and much error but the piece I made screws on a little crooked and does not feel mechanically sound. One thing I have noticed about this thread type is that it is not a singe continuous revolution, its three separate half revolutions. Does anyone here know how to make this thread type correctly and or what the actual thread type/size is?