I am looking for references/applications. However note that I am aware of the link between semigroups and CS languages (namely via finite automata). (Does this literature on semiautomata ever look at "group-automata"?)
I have seen one paper many years ago that might come close, that converts TM transition tables into a binary operation, possibly sometimes a group in some cases, conceivably based on some kind of symmetry in the TM state table. It didn't explore that in particular, but also didn't rule it out.
Also, in particular, regarding the large body of math research on classification of finite groups, does or could it have any meaning or interpretation in TCS? What is the "algorithmic lens" view of this massive edifice of mathematical research? What is it "saying" about a possible hidden structure in computation?
This question is partly inspired by some other notes e.g.: