Size of conjugacy class of subgroups equals index of normalizer

Statement

Let $G$ be a group and $H$ be a subgroup of $G$. Let ${\mathcal {C}}$ denote the conjugacy class of subgroups of $H$ in $G$, i.e., the set of all subgroups of $G$ that are Conjugate subgroups (?) to $H$ (note that $H\in {\mathcal {C}}$). Then, there is a bijection:

$G/N_{G}(H)\leftrightarrow {\mathcal {C}}$

where $N_{G}(H)$ is the Normalizer (?) of $H$ in $G$ and $G/N_{G}(H)$ denotes the coset space. In particular:

$[G:N_{G}(H)]=|{\mathcal {C}}|$

In other words, the size of the conjugacy class of subgroups ${\mathcal {C}}$ equals the index of $N_{G}(H)$ in $G$.

Facts used

1. Group acts as automorphisms by conjugation
2. Fundamental theorem of group actions