Let $\omega(n)$ be the number of prime factors of $n$ and let $$a_n = \begin{cases} 1 & \omega(n) < (3/2) \log \log n \\ -1 & \omega(n) > (3/2) \log \log n \end{cases}. $$
By Erdős–Kac, $\frac{1}{N} \sum_{n \leq N} a_n$ is very close to $1$.
On the other hand, by Erdős–Kac applied to $n/q$, $\frac{1}{N/q} \sum_{n \leq N \colon q\mid n} a_n $ is very close to $-1$ as long as $\omega(q) > (1/2 + \epsilon) \log \log n$.
So the difference $\frac{1}{N/q} \sum_{n \leq N \colon q\mid n} a_n - \frac{1}{N} \sum_{n\leq N }a_n $ is close to $-2$ as long as $\omega(q) > (1/2 + \epsilon) \log \log n$.
By Erdős–Kac one last time, this property holds for almost all $q \leq Q$ as long as $Q > e^{ \sqrt{ \log n }}$.