Robust Exact Variance of the Sample Variance for a Normal Population

Let $X_1, \ldots, X_n$ be iid $N(\mu, \sigma^2)$ and define the sample variance $S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2$.

(a) Identify the distribution of $(n-1)S^2 / \sigma^2$ and use it to derive an exact expression for $\operatorname{Var}(S^2)$.

(b) Evaluate $\operatorname{Var}(S^2)$ when $n = 10$ and $\sigma^2 = 3$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.