If $G$ is a finite group, $H\leq G$ a subgroup, and $n$ divides $|G|$, then $n$ divides $[G:N_{G}(H)]$ times the number of elements $g\in G$ for which $|\langle H,g\rangle|$ is a divisor of $n$.
This appears as Theorem 6.1 of van der Lee - Elementary proof of a theorem of Hawkes, Isaacs, and Özaydin. I was not able to find a source myself, and I wondered whether the result was new. Any references for this result?
(When $H=1$, this is just Frobenius' theorem, by which the number of $g\in G$ with $g^{n}=1$ is a multiple of $n$, for $n\mid |G|$.)