Prove that no group $G$ satisfies $G/Z \approx Q_8$ where $Z$ is centre of $G$

Question

Prove that no group $G$ satisfies $G/Z \approx Q_8$ where $Z$ is centre of $G$

My initial thoughts on this were the following:

1. The image of a group under an isomorphism with a trivial centre must also have a trivial centre.
2. $Q_8$ has a non trivial centre, $ \{ \pm 1 \} $.

So it would be nice to claim that $G/Z$ has a trivial centre and so we are done. However I do not think this is the case, e.g. $D_8 /Z \approx D_4 $ and the centre of $D_4$ is nontrivial (contains rotation element).

Firstly is my above claims correct, and what ideas should I consider to solve this problem?