From the conclusion of Weyl's Symmetry —
One example of Weyl's "structure-endowed entity" is a partition of a six-element set into three disjoint two-element sets– for instance, the partition of the six faces of a cube into three pairs of opposite faces.
The automorphism group of this faces-partition contains an order-8 subgroup that is isomorphic to the abstract group C2×C2×C2 of order eight–
The action of Klein's simple group of order 168 on the Cayley diagram of C2×C2×C2 in yesterday's post furnishes an example of Weyl's statement that
"… one may ask with respect to a given abstract group: What is the group of its automorphisms…?"