This is Exercise 191 of Rose's "A Course in Group Theory".
The Details:
From p. 38 ibid.,
Definition: Let $H\le G$ and define $$H_G=\bigcap_{g\in G}(g^{-1}Hg).$$ [. . .] It is called the core (or normal interior) of $H$ in $G$.
From p. 56 ibid.,
Definition: [Here] $\varpi$ is a set of prime numbers. [. . .] A positive integer $n$ is said to be a $\varpi$-number if every prime divisor of $n$ belongs to $\varpi$. [. . .] We say $G$ is a $\varpi$-group if $\lvert G\rvert$ is a $\varpi$-number.
The Question:
Let $H\le G$ with $|G:H|=n<\infty$. Let $\varpi$ be the set of all primes not exceeding $n$. Then $G/H_G$ is a finite $\varpi$-group.
This is on page 75.
Context:
My progress in Rose's book and my memory of the preceding material have both been stunted by this question. I have made little if any progress on it.
I think the group ${\rm Inn}(G)$ of inner automorphisms of $G$ might play an important role here.
An Example:
Let $\varpi=\{ 2,3\}$, $\Bbb Z_2\cong H\le G\cong \Bbb Z_6$. Then $n=[G:H]=3<\infty$. Suppose $G$ is generated by $a$. Then $H=\langle a^3\rangle$. We have
$$\begin{align} H_G&=\bigcap_{g\in G}(g^{-1}Hg)\\ &=H\cap\{ a^{-1}ha:h\in H\}\\ &\cap\{ a^{-2}ha^2:h\in H\}\\ &\cap\{ a^{-3}ha^3:h\in H\}\\ &\cap\{ a^{-4}ha^4:h\in H\}\\ &\cap\{ a^{-5}ha^5:h\in H\}\\ &=\bigcap_{i=1}^6\langle a^3\rangle\\ &=H. \end{align}$$
Then $G/H_G\cong G/H\cong \Bbb Z_3,$ which is indeed a finite $\varpi$-group.
Perhaps the choice of $G$ and $H$ is not helpful as they are both finite and cyclic (and hence abelian).
Something more instructive might be found in $G=S_3$, say, as that is the smallest nonabelian group, or in $2\Bbb Z\le \Bbb Z$, since they're infinite (although then $\varpi=\{2\}$, which doesn't tell us much).
I suspect that, with more time, I'd be able to answer this myself, but I've already spent "too long" on it for a recreational/peripheral exercise. I get the feeling that I'm missing some important conceptual aspect of the problem though.
The type of answer I'm hoping for is a strong hint or a full proof (to address what it is I don't understand yet).
Please help :)