Generalizing Lehmer’s totient problem

An important unsolved question in number theory is 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 Lehmer's totient problem is also proposed.