This question probably has a very simple answer!
I'm trying to understand the proof of the following result from Dummit and Foote, 3ed:
Here is the proposition referenced:
I don't understand the part where Proposition 13 is applied "with $N_G(H)$ playing the role of $G$". Wouldn't this only give me that $N_G(H)/C_{N_G(H)}(H)$ is isomorphic to a subgroup of $\text{Aut}(H)$? How does $C_G(H)$ appear?
Thanks for any help.