Sets of Operations on Set of 3 Elements/Automorphism Group of B/Isomorphism Classes

Theorem
Let $S = \set {a, b, c}$ be a set with $3$ elements.

Let $\BB$ be the set of all operations $\circ$ on $S$ such that the group of automorphisms of $\struct {S, \circ}$ forms the set $\set {I_S, \tuple {a, b, c}, \tuple {a, c, b} }$, where $I_S$ is the identity mapping on $S$.

Let $\oplus \in \BB$.

Then the isomorphism class of $\oplus$ consists of $\oplus$ and exactly one other operation $\otimes$ on $S$ such that $\otimes \in \BB$.

That is, the elements of $\BB$ are in doubleton isomorphism classes each of which is a subset of $\BB$.