My problem is the following:
Given a group $G$ and a normal subgroup $N$ with $|N| = 3$ which is not contained in the center of $G$. Show that $G$ contains a subgroup $H$ with $[G:H] = 2$.
Attempted Solution: We simply leverage the NC Theorem to obtain the desired subgroup. By the assumptions on our subgroup $N$, we have that $$N_G(N) / C_G(N) = G / C_G(N) \le Aut(N)$$ and since $|N| = 3 \Rightarrow N \cong \mathbb{Z}_3$, which further gives us that $Aut(N) \cong \mathbb{Z}_2$. Combining these results, we have that $G / C_G(N) \cong \mathbb{Z}_2$ and since $C_G(N) \lhd G$, we have the desired subgroup.