Let G=Sym(n) and \theta \in Hom(C, G) where C is a cyclic group. Since C is cyclic \gamma (Ker \theta)=Ker \theta, where \gamma \in Aut(C) and the generators of \theta(C) in G have the same cycle type. Therefore the map \tilde \theta :N_{G}(\theta (C))\longrightarrow Aut(C/Ker \theta) is an isomorphism.

Can u explain the last sentence?

And Is this true for all groups? or is there a special group?