Given $\displaystyle K$ is a normal subgroup of a group $\displaystyle G$. The map $\displaystyle \psi : G$ --> $\displaystyle G/K $ by $\displaystyle \psi(g) = \phi(g)K$ for $\displaystyle g \in G$ is a homomorphism and $\displaystyle \phi \in Aut(G)$.

Show $\displaystyle \psi(K) = (\phi(K)K)/K$.