Let $G$ be a finite group, and let $P$ be a finitely generated group.
Consider the number $$n=\#Hom\_{Grp}(P,G).$$
It is known (see [Number of solutions to equations in finite groups](https://mathoverflow.net/questions/198306/number-of-solutions-to-equations-in-finite-groups)) that under relative mild assumptions on $P$ the number $n$ is divisible by $|G|$. I would like to ask if the following is also true:



> 
> **QUESTION.** Are all the prime divisors of $n$ also prime divisors of $|G|$?
> 
> 
> 


More generally, is it true that if we consider ${all}$ finitely presented groups $P$, the collection of numbers $$n\_P=\# Hom\_{Grp}(P,G)$$ has only finitely many prime divisors (for the given group $G$)? It is true, for quite simple reasons, in case the group $G$ is abelian. I do not know however what happens for a general $G$.



Number of homomorphism, or number of solution to equations, in finite groups
There are $70$ homomorphisms from $\mathbb{Z}\times\mathbb{Z}$ to the dihedral group of order $14$.


