The Bombieri-Vinogradov Theorem states that given $A>0$, for $\frac{\sqrt{x}}{\left(\log x\right)^{A}}\leq Q\leq\sqrt{x},$  we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\text{ mod q}\\
(a,q)=1
\end{array}}\left|\psi(y;q,a)-\frac{y}{\phi(q)}\right|\ll x^{\frac{1}{2}}Q(\log(x))^5.$$ 

I was wondering what happens if a restriction is put on the $q$ so that they are all divisible by some smaller integer $k$. Are there any non-trivial bounds on the average over $q$ divisible by $k$? Specifically, suppose that $k\leq Q^{1-\epsilon}.$ Is it true that $$\sum_{\begin{array}{c}
q\leq Q\\
k|q
\end{array}}\max_{y\leq x}\max_{\begin{array}{c}
a\text{ mod q}\\
(a,q)=1
\end{array}}\left|\psi(y;q,a)-\frac{y}{\phi(q)}\right|\ll_{\epsilon}\frac{1}{k} x^\frac{1}{2}Q(\log(x))^5.$$ 

Presumably issues arise when $k$ is very close to $Q$, as above, but perhaps this holds in a smaller range of $k$.  I would be interested if it holds for $k<Q^\delta$ for some $\delta>0$.   Any references would be greatly appreciated.

Thanks for your help,