Foreward: This post not a puzzle in itself, but an invitation for you to show off your puzzle-making skill regarding a given topic. Although the tag description of "puzzle-creation" seems to discourage its use in this manner, a search of its posts shows a different history and user consensus, e.g. mostly recently here. Let me know if there's some part of this ambiguity that needs to get sorted out on the Meta site or something.
It's not exactly a secret that many brainteasers can be reformulated (often as part of a Eureka moment) into a precise mathematical problem. See, for example, any Lights Out-type problem, any Konigsberg bridge-type problem, any unique-up-to-symmetry counting problem solved by Burnside's lemma, any problem solvable by traditional combinatorial game theory, or even some of my own recent puzzles. I think math makes a great inspiration for new puzzles, and creating puzzles can offer unique interpretations of mathematical results.
It's a known fact that $S_6$ (group of permutations of six objects) has exactly one "exceptional" automorphism, up to relabeling of the objects. I've always found it interesting that this fact about $S_n$ is unique to $n=6$. There are plenty of problems which sneak permutation theory or group theory into riddles (an example of the first: the 100 prisoners problem. An example of the second: Rubik's puzzles, 15-piece puzzles, or insolubility thereof. Some card-shuffling problems can fall into one or the other).
What would be a good puzzle whose statement involves as little math as possible, but whose solution intimately involves this exceptional automorphism?