I have the sample $Y_{i}\overset{i.i.d}{\sim} N(0,\sigma_{y}^{2})$ of a population with normal distribution with mean $0$ and variance $\sigma_{y}^{2}$ unknow, also I could prove that $\frac{1}{n}\left(\frac{1}{n}\sum_{i=1}^{n}Y_{i}^{2}-\sigma_{y}^{2}\right)\overset{D}{\to}N(0,2\sigma^{4})$ using the CLT, but my question is:

Question: How can I find a function $f(t)$ such that the variance of the limiting distribution does not depend on the parameter of interest $\sigma_{y}^{2}$ in the sense of Delta'method: $$\sqrt{n}\left(f\left(\frac{1}{n}\sum_{i=1}^{n}Y_{n}\right)-f(\mathbb{E}X_{i})\right)\overset{D}{\to} N(0,\sigma^{2}(\mu)f'(\mu)^{2}) , \quad Y_{i}\overset{i.i.d}{\sim}N(\mu,\sigma^{2})$$ Foe example, I could see that $f$ might satisfy $2(f'(\sigma^{2}))^{2}\sigma^{4}$ shouldn't depend of$\sigma^{2}$.

I have the sample $Y_{i}\overset{i.i.d}{\sim} N(0,\sigma_{y}^{2})$ of a population with normal distribution with mean $0$ and variance $\sigma_{y}^{2}$ unknown. I could also prove that $\frac{1}{n}\left(\frac{1}{n}\sum_{i=1}^{n}Y_{i}^{2}-\sigma_{y}^{2}\right)\overset{D}{\to}N(0,2\sigma^{4})$ using the CLT, but my question is:

Question: How can I find a function $f(t)$ such that the variance of the limiting distribution does not depend on the parameter of interest $\sigma_{y}^{2}$ in the sense of the Delta method: $$\sqrt{n}\left(f\left(\frac{1}{n}\sum_{i=1}^{n}Y_{n}\right)-f(\mathbb{E}X_{i})\right)\overset{D}{\to} N(0,\sigma^{2}(\mu)f'(\mu)^{2}) , \quad Y_{i}\overset{i.i.d}{\sim}N(\mu,\sigma^{2})$$

For example, I could see that $f$ might satisfy $2(f'(\sigma^{2}))^{2}\sigma^{4}$ shouldn't depend on $\sigma^{2}$.