This question is not about some kind of mathematical permutation of notes, but a perceptual categorization, mostly related to jazz and improvisation.
There's a video of Jacob Collier answering what was his favorite chord at the moment and it was the chord with the following voicing: F,D♭,A,B,E,G♯. It would be an F with ♯11, M7, ♯5, ♯9, but he says "it's basically an F major". Likewise, Rick Beato also categorizes augmented chords in the same group as major chords in some of his videos.
For these people, sometimes an augmented chord is not an important entity that deserves a new category. In some sense, it shares many features with the category "major chord", so it's inside the same group. I can think of some "fundamental" categories: major, minor, dominant, diminished. (could a half-diminished always be seen as an inversion of a minor?)
How do you approach this type of categorization? In what sense an augmented chord IS a major chord? What is it being taken into account?
Just some thoughts:
It's possible to see the existence of accessory/embellishment notes in some chords. CM7, CM13, C6add9 or even CM♯11 can be seen as C triads with cherries on top. They are treated in similar ways when improvising. Also, Cm7, Cm9, Cm(M7) are treated in a similar fashion (sometimes musicians even play the 7M over a Cm7 and vice-versa and it can sound good). On the other hand, the addition of a minor 7th on a C makes it fundamentally different from a CM7. You don't land on a 7M over a dominant chord. The C7 behaves and carries different expectations than the CM7.
Some categories might also be hidden inside others in another way. All possible triads can be mathematically defined, but the chord with the notes C-F♯-G for example, (which could be called a "lydian triad") is heard as something major in the lydian world. The ear gravitates towards the perfect fifth (it's not a diminished thing with an added perfect 5th, but a triad with a perfect 5th and an added ♯4), and it fills a major third. It's not exactly a new entity.
In this sense, I would have the categories: minor, major, dominant, diminished. And I'd have some doubts about half-diminished and augmented. A better systematization of my thoughts is needed and it might have limited application, but I'm interested in hearing other definitions and opinions.