I asked this question on MSE here
FromThis question was inspired by: The influence of conjugacy class sizes on the structure of finite groups.
My question is as follows: Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically? Could you please provide references for this?
Clarification of the question: we know the impact of conjugasyconjugacy classes in classification the classification of finite groups, but however there are open questions dose not have answers yetremain, so my main gaolgoal is to find a way to connect category of finittheory with questions related to finite groups and conjugacy classes categorically, to solve such questionsthem.