Let $X$ and $Y$ be two quandles and $f: X \rightarrow Y$ be a quandle homomorphism. Then we can define a map $\bar f: Inn(X) \rightarrow Inn(Y)$ as $\bar f(S_a)=S_{f(a)}$, where $a \in X$. Then $\bar f$ may not be a group homomorphism. But I am not able to construct such example.

I have posted this question on Math Stack Exchange, but have not got any reply.The link is here https://math.stackexchange.com/questions/2859327/quandle-homomorphism-does-not-always-induces-group-homomorphim-on-inner-automorp.

Any suggestions or hints would be of great help.