Dummit and Foote leave this as an "easy exercise for verification" and I would like to verify that I am indeed reasoning correctly.
Proof Take an arbitrary element $kH \in K/H$, and let $G/H$ act on $K/H$ by conjugation. $(gH)(kH)(gH)^{-1} = (gH)(kH)(H^{-1}g^{-1}) = gh_1 kh_2 h_3^{-1} g^{-1} = gh_1kh_4g^{-1}$ But $H\le K$, and so $h_1 \in H$ is also in $K$, then $h_1 kh_4 \in K$, and $K$ is normal in $G$, so $gKg^{-1} = K$, and thus $(gH)(kH)(gH)^{-1} \in K$.
My Question
How do I recover my coset $kH \in K/H$? I have made an error along the way, and can't see where my error lies.