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

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 $$\sum_{\begin{array}{c}
q\leq Q\\
k|q
\end{array}}\max_{y\leq x}\max_{\begin{array}{c}
a\text{ mod q}\\
(a,q)=1
\end{array}}\left|\psi(y;q,a)-\frac{y}{\phi(q)}\right|\ll_{\epsilon}\frac{1}{k}\frac{x}{\left(\log x\right)^{A}}.$$ 

Presumably issues arise when $k$ is very close to $Q$, but perhaps this holds in a smaller range of $k$.  Any references would be greatly appreciated.

Thanks for your help,