As a project with a certain 4-year-old relative of mine, we constructed the proof I described before that the outer vertices of George Hart’s 12-Card Star form a 4!-torsor. (I guess I didn’t say it that way before, but it’s true!) Here’s our proof:
You can get any permutation of these colors in our Star by exactly one symmetry taking outer vertices to outer vertices. The “exactly one” in this isomorphism is what makes the set of outer vertices a 4!-torsor rather than just a 4!-set.
Here’s what it looks like when you put three pieces together, from both sides: