Invariance moves the MLE but not the UMVUE

Let $X_1,\ldots,X_n$ be independent $\operatorname{Bernoulli}(p)$ observations, $0\le p\le1$, $n\ge2$, and let $g:[0,1]\to\mathbb R$ be any function, not necessarily injective. Let $\hat p$ be a maximizer of the Bernoulli likelihood. Prove that $g(\hat p)$ maximizes the induced likelihood for the target $\tau=g(p)$. State the induced-likelihood definition you use so that the claim remains valid when $g$ is not one-to-one. Take $g(p)=p(1-p)$. Write the MLE of $p(1-p)$ explicitly in terms of $\bar X$. Then compute $\mathbb E_p[\bar X(1-\bar X)]$ and produce an unbiased estimator of $p(1-p)$. Explain why invariance does not force that unbiased estimator to equal the MLE, and why a UMVUE need not be a function $g(\hat p)$ of the MLE of $p$. Do not treat this as a test of $H_0:p=p_0$, and do not Rao–Blackwellize a Poisson zero count.

If $L(\hat\theta)\ge L(\theta)$ for every $\theta$, then the supremum of $L$ over each $g$-fiber is attained at a fiber containing $\hat\theta$.