Which groups of order 16 can occur as a central extension of $Q_8$, the group of quaternions, by $\mathbb{Z}/2$?
I know that the direct product $Q_8\times \mathbb{Z}/2$ and the semidirect product $\mathbb{Z}/4\rtimes \mathbb{Z}/4$ can occur (see this reference), but there should also be (at least) one other group as $H^2(Q_8;\mathbb{Z}/2)=\mathbb{Z}/2\oplus\mathbb{Z}/2$ and the pullback map $H^2(Q_8;\mathbb{Z}/2)\rightarrow H^2(Z(Q_8);\mathbb{Z}/2)$ is non-zero!