The popularity of values of Euler’s function

For each positive integer $m$, let $N(m)$ denote the number of $\varphi$-preimages of $m$, where $\varphi$ is Euler’s totient function. For example, $N(12) = 6$, corresponding to the six preimages 13, 21, 26, 28, 36, and 42. We discuss several statistical questions concerning $N(m)$ — for instance, its average size, its maximal order, and the typical size of
$N(\varphi(k))$ as $k$ varies.