Generalizing Lehmer’s totient problem

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