Excuse me? I did not "slam" the question anywhere. This is purely your interpretation and therefore your problem. I factually stated that reading original research papers in order to learn is not the best way of going about things. Your suggestion to direct to Wilczek's thesis (not paper) is counterproductive as Wilczek's thesis is also original research even if it has been awarded a Nobel prize. Reading Wilczek's thesis is not the best way of going about learning about asymptotic freedom. The best way is to pick up a modern textbook on quantum field theory (e.g., Schwartz).

AFAIK there is no such thing. Any theory that embeds SU(3) within a larger symmetry would amount to some kind of supersymmetric model; that's what "supersymmetry" means.

"Supersymmetry" typically refers to models with an extended space-time symmetry with anti-commuting generators, not to models that embed the SM gauge groups in a larger gauge group, which are typically unification models where all SM gauge groups are subgroups of a larger gauge group at higher energies, e.g., SO(10). The larger gauge symmetry is then broken at some high scale and the couplings of the remaining symmetries separately run down to the scales that we can observe.

How about preon models (those with subquarks). Do they automatically have larger symmetry group? Can someone share one preon model that is still Su(3) only?

in the references inside it are the following papers: http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-2310.pdf
"It would be premature to insist, for example, that presently established ideas of gauge theories are sufficient for fully explaining the interactions of the new hypothetical building blocks. In fact, the correct dynamics at very short distances may be radically different, and is likely to involve some
entirely new principles. However, when viewed at present energies and distances, in which quarks, leptons and ordinary gauge bosons are "point-
like", it should somehow reproduce currently accepted theories such as ' SU(2) x U(1) and QCD"

https://arxiv.org/pdf/hep-ph/9709227.pdf "Higgs Pain? Take a Preon!"
"Maybe normal QCD is nothing but the ”long-range” tail of the hyper-QCD that acts between preons, reaching out from the coloured quarks, but not from the leptons."

https://arxiv.org/ftp/hep-ph/papers/0411/0411313.pdf
"Why quarks cannot be fundamental particles
Many reasons why quarks should be considered composite particles are found in the book Preons by D'Souza and Kalman. One reason not found in the book is that all the quarks except for the u quark decay. The electron and the electron neutrino do not decay. A model of fundamental particles based upon the weak charge is presented."

Arxiv has many references as well. https://arxiv.org/pdf/1307.6133.pdf
"In a model in which leptons, quarks, and the recently introduced hyperquarks are built up from two fundamental spin-1 2 preons, the standard model weak gauge bosons emerge as preon bound states. In addition, the model predicts a host of new composite gauge bosons, in particular those responsible for hyperquark and proton decay. Their presence entails a left-right symmetric extension of the standard model weak interactions and a scheme for a partial and grand uniﬁcation of nongravitational interactions based on respectively the eﬀective gauge groups SU(6)P and SU(9)G."

I can't find the book above and others to see more details. What I'd like to know is that If there are subquarks.. would the main symmetry group still be SU(3) or would SU(3) just be residual effect? And most importantly.. does the higher symmetry group occur at low energy below the electroweak scale or even below at the low quark-gluon plasma scale. Here we are making distinction to grand unified theory which has GUT energy scale. I'm asking about preon model where the energy scale is same as the SU(3) scale only.

So bottom line. Should all preon subquark model extend the symmetry group SU(3) or do they still use this even for the interactions among the preons inside the quark?

It would have been a better idea to read up on the relevant literature before starting this thread.

The answer is "it depends"--it depends on the model. "Preon" does not name a specific model; it just names a general idea, that there should be "sub-particles" of which quarks and leptons are composed. A grand unified theory such as the SU(5) GUT, in which quarks and leptons are particular combinations of the underlying SU(5) particles, would meet this definition, so it's an example of a "preon" model. But there might be other "preon" models that work differently. And since none of this can be tested experimentally, it's all just speculation anyway.

In short, the questions you are asking don't really have definite answers. That being the case, this thread is closed.