Is the maximum likelihood estimator always a sufficient statistic? Here's an example of what I am asking :

$X_1,\ldots,X_n $ i.i.d $N(\phi , 1)$ where $\phi \in \mathbb{R}$. Let $\gamma = P(X_1\leq 1)$.
Give a sufficient statistic of $\gamma$.

This question is part of a midterm exam I took on april.
I see that $\gamma_{m\ell} = \Phi(1 - \phi_{m\ell}))$, where  $\Phi$
is the standard normal distribution function. I know that in this case,   $\phi_{m\ell} = n^{-1}\sum(X_i^2)$ which is a sufficient statistic. Does that make $\gamma_{m\ell}$ one also ?
Thank you for your answers.
 A: Several points of confusion are in this question:


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*The subject line says "Is the maximum likelihood estimator always a sufficient statistic?". That would leave the impression that that is what this question is about. The short answer is "no". More on that below . . .

*Then it says "Here's an example of what I am asking :" and then presents the following problem:       $X_1,\ldots,X_n $ i.i.d $N(\varphi , 1)$ where $\varphi \in \mathbb{R}$. Let $\gamma = P(X_1\leq 1)$.      Give a sufficient statistic of $\gamma$. To approach that question by thinking about MLEs might make sense if the MLE were always a sufficient statistic. However, it is easy to show that $X_1+\cdots+X_n$ is sufficient for $\gamma$ without going into that. I answered that question today, here. I briefly considered closing this present question as a duplicate of that. 
Following up on the comment by "spaceisdarkgreen": The following answer deals with a case in which the MLE is not a sufficient statistic:
https://stats.stackexchange.com/questions/174117/maximum-likelihood-estimator-of-location-parameter-of-cauchy-distribution
That answer gives a numerical method for finding the MLE. How do we know it is not a sufficient statistic? That is dealt with in Bernard Lindgren's Statistical Theory, 4th edition, and right now I can't find it with the help of the index. On one occasion I told the author of that book that I couldn't find it, and he pointed out where it is, and I've forgotten, so definitely this is a deficiency in the index. He died shortly after that, so I won't be able to ask him again. However, it is commonplace to find it asserted, even if unusual to find it proved, that the complete set of all $n$ order statistics from a sample of size $n$ is the coarsest you can get with the Cauchy family. For any i.i.d. sample, the complete set of order statistics is sufficient, but for many it is not minimal. For this one it is minimal. So this is a case in which the MLE is not a sufficient statistic. I think there are simpler examples, but I don't have one at the tip of my tongue. Romano & Siegel's book Counterexamples in Probability and Statistics might have one.
