  An important unsolved question in number theory is the Lehmer's totient
problem that asks whether there exists any composite number $n$ such that
$\varphi(n)\mid n-1$, where $\varphi$ is the Euler's totient function. It is
known that if any such $n$ exists, it must be odd, square-free, greater that
$10^{30}$, and divisible by at least $15$ distinct primes. Such a number must
be also a Carmichael number.
  In this short note, we discuss a group-theoretical analogous problem
involving the function that counts the number of automorphisms of a finite
group. Another way to generalize the Lehmer's totient problem is also proposed.
