I describe below an involutive quandle which satisfies the equivalent conditions of Proposition 5.2, equivalently, condition (5.2), of my paper On core quandles of groups, but which cannot be embedded in Conj(G) for any group G. In view of the second statement of Proposition 6.2 of that paper, such a quandle also cannot be embedded in Core(H) for any group H. Hence, this example gives negative answers to Questions 5.3 and 5.5 of that paper.
Let Q be an 8-element set with elements named xi, yi, zij (i, j ∈ {0, 1}). Define a binary operation ◁ on Q as follows: Let each of the subsets {x0, x1}, {y0, y1}, {z00, z01, z10, z11}, act trivially on itself under ◁. Let the action on {x0, x1} of every element not in that set interchange x0 and x1, and similarly let the action on {y0, y1} of every element not in that set interchange y0 and y1. Finally, let the action of each of the x's on the z's interchange 0 and 1 in the first subscript, leaving the second subscript unchanged, and let the action of each of the y's on the z's interchange 0 and 1 in the second subscript, leaving the first subscript unchanged. It is straightforward to verify that this is an involutory quandle structure, and satisfies condition (5.2).
Suppose, now, that this quandle Q were a subquandle of Conj(G) for some group G. We may assume without loss of generality that G is generated by the elements of Q.
Note that since x0 and x1 have the same action by ◁ on Q, which generates G as a group, they must have the same action by conjugation on all elements of G, hence they must differ in G by a central element a; and since the conjugation automorphisms that take x0 to x1 also take x1 = x0 a to x0, we must have a2 = 1. Letting w denote any of the yi or zij, the relation w−1xi w = xi a can be rewritten w = xi−1w xi a, whence, using the fact that a2 = 1, we get xi−1w xi = wa. Letting w = yi in this relation, we see that y0 and y1 differ by a factor of a. Similarly, letting w = zij, we see that z0j and z1j differ by a factor of a.
But knowing that y0 and y1 differ by a factor of a, we can repeat the observations of the above paragraph with y0 and y1 in place of x0 and x1, this time concluding (in view of the different action of the yi on the zij ) that zi0 and zi1 differ by a. Combining these results, we conclude that z00 and z11 differ by a factor of a2 = 1, i.e., are equal, contradicting our original description of Q, and thus showing that Q cannot, as we had assumed, be embedded in a quandle Conj(G).
Remark: A key property of Q that made the above reasoning possible is that the elements of each orbit of Q all induce the same permutation of Q, i.e., that Q satisfies the identity (a ◁ b) ◁ c = b ◁ c. Quandles satisfying this identity are studied in
Victoria Lebed and Arnaud Mortier, Abelian quandles and quandles with abelian structure group, J. Pure Appl. Algebra, 225 (2021), no. 1, 106474, 22 pp., MR4116819,
where they are called abelian quandles.
I do not currently have plans of publishing the above example.