This question will likely be closed, but I'll try to give the OP some references before it is. First, let me answer [the OP's question in the comments][1], regarding why this is vague and why the question will likely be closed: 1. It's full of typos (I will try to fix them, but their presence suggests the OP didn't put much effort into writing the question). 2. It was cross-posted simultaneously. It would have been better to post to MSE, engage with answers there for a week, think deeply, and only post to MO if there was still confusion that couldn't be answered at the level of MO. 3. The MSE version has an extra part of the question that shows the OP is a relative novice, putting this far below research level. 4. It's vague, because actually there is a HUGE research area on this connection, as you can see below. The short answer to the question is "yes, of course." There are SO MANY ways to get at finite groups and conjugacy classes using category theory. Here are just a few off the top of my head. 1. Fusion systems. https://web.mat.bham.ac.uk/C.W.Parker/Fusion/fusion-intro.pdf https://www.math.univ-paris13.fr/~bobol/ako.pdf https://maths.nuigalway.ie/~park/papers/intro-fusion-systems.pdf 2. Mackey functors https://ncatlab.org/nlab/show/Mackey+functor https://people.math.rochester.edu/faculty/doug/otherpapers/WebbMF.pdf Relatedly, there is a history of Japanese mathematicians using Mackey and Tambara functors on questions related to finite simple groups. I learned about this from [Hiroyuki Nakaoka][2]. I recommend checking out the references in [his early papers][3] on the subject. 3. Transfer systems, [indexing systems][4], and homotopical combinatorics. Getting at the poset of conjugacy data. https://www.ams.org/notices/202402/rnoti-p260.pdf https://kyleormsby.github.io/posts/2021/09/homotopical-combinatorics/ 4. The following book takes a categorical lens in some places: http://homepages.math.uic.edu/~smiths/book.pdf Based on the MSE thread, I think the OP might like even more basic ways of getting at conjugacy data using categories, and will get such answers over there. I wanted to post about some higher level approaches that are active research areas at the moment. [1]: https://mathoverflow.net/questions/467524/is-there-a-way-to-study-the-relationship-between-the-category-of-finite-groups-a/467535#comment1213696_467524 [2]: https://arxiv.org/search/math?searchtype=author&query=Nakaoka,%20H [3]: https://arxiv.org/pdf/1010.0812.pdf [4]: https://arxiv.org/pdf/2212.11250.pdf