(1) Let G=Sym(n) and $\displaystyle \theta \in Hom(C, G)$ where C is a cyclic group. Then $\displaystyle \gamma (Ker \theta)=Ker \theta $ , where $\displaystyle \gamma \in Aut(G)$.

(2) Can you explain , why this (1) breaks down for a non-cyclic abelian group? Pls give an example.