# abstract algebra help

Given $K$ is a normal subgroup of a group $G$. The map $\psi : G$ --> $G/K$ by $\psi(g) = \phi(g)K$ for $g \in G$ is a homomorphism and $\phi \in Aut(G)$.
Show $\psi(K) = (\phi(K)K)/K$.
Since $K$ is a subgroup of $G$, $\psi(K)$ is a subgroup of $G/K$. So for every $x \in \psi(K), x \in G/K$, and then $x = gK$ for some $g \in G$.
I need to show that $\psi(K)$ and $(\phi(K)K)/K$ are subsets of each other, but I'm stuck in here, did I start this right?