= Solution
For observations $X_1,\ldots,X_n$, the <log-likelihood> is, up to an additive constant,
$$
\ell(\sigma)=-n\log\sigma-\frac1{2\sigma^2}\sum_iX_i^2.
$$
Its unique maximum is
$$
\widehat\sigma_{\rm MLE}=\sqrt{\frac1n\sum_iX_i^2}.
$$
Now $\mathbb E[X_i^2]=\sigma^2$ and, using $\mathbb E[Z^4]=3$, $\operatorname{Var}(X_i^2)=2\sigma^4$. The <central limit theorem> followed by the <delta method> for $g(x)=\sqrt x$ gives
$$
\boxed{\sqrt n(\widehat\sigma_{\rm MLE}-\sigma)
\xrightarrow dN(0,\sigma^2/2)}.
$$
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