This is an exercise I am stuck on.
Given an IID sample $X_1, \dots, X_n \sim N(\mu, \sigma^2)$ with $\mu \ne 0$ let
$$S_n^2 = \frac{1}{n} \sum_{i = 1}^n ( X_i - \overline{X_n})^2$$
be the sample variance and $\overline{X_n}$ the sample mean. I am tasked in finding the asymptotic distribution of $S_n^2$ using the second order delta method. This says that given a continuous and doubly differentiable function $\phi$ with $\phi'(\theta) = 0$ and an estimator $T_n$ of a parameter $\theta$ such that $$\sqrt{n} (T_n - \theta) \rightarrow_d T $$ where $T$ is some probability density function then $$\sqrt{n} (\phi (T_n) - \phi(\theta)) \rightarrow_d \frac{1}{2} \phi''(\theta) T^2 $$.
In my exercise I think I have to utilize this method taking as $T_n$ the sample mean, the problem I run into is that I don't know how to define the function $\phi$. I know that
$$ \sum_{i = 1}^n ( X_i - \overline{X_n})^2 = \sum_{i = 1}^n X_i^2-n \overline{X}_n$$ but this does not help me in defining $\phi$. Moreover as a hint I am given to work with $$\frac{S^2_n}{\overline{X_n}}$$ and I can't see why this would help. Any ideas?
EDIT: My theorem was wrong as pointed out by Chaconne in his answer, I correct it for the sake of clarity. I am still searching for the appropriate $\phi$ to utilize to apply the delta method to my exercise.