Abstract:We study the algebra of all binary operations on a finite set, with composition defined as follows: for two binary operations A and B, their composition at an ordered pair x and y is B applied to the pair consisting of A(x,y) and A(y,x). A canonical family of automorphisms arises by conjugation with permutations of the underlying set. We characterize how these permutation-induced automorphisms act on several natural invariant subsets and show that the permutation they induce on constant operations determines every operation’s values on the diagonal. Beyond conjugation by permutations, we identify additional symmetries that are not conjugate to these permutation conjugations, and we show that conjugation by permutations need not be surjective onto the full automorphism group of the algebra under the given composition. For small underlying sets we obtain complete descriptions: when the set has two elements the automorphism group is isomorphic to the two-element symmetric group, and when the set has three elements it is isomorphic to the direct product of the three-element symmetric group with a cyclic group of order two. The paper concludes with algorithmic remarks and open problems toward a full classification of the automorphism group.