Embedding the generalised quaternion group into a general linear group

Question

It's known that there are four non-abelian groups with cyclic subgroup of index $2$. Those groups are the dihedral group $D_{2^n}$, generalised quaternion group $Q_{2^n}$, modular-maximal group $M_{2^n}$ and semidihedral group $SD_{2^n}$. It's also known that other $3$ groups $D_{2^n}$, $M_{2^n}$, $SD_{2^n}$ are embedded in $\operatorname{GL}(2, \mathbb{Z}_{2^{n-1}})$, so I tried to find such an embedding for $Q_{2^n}$ but wasn't able to find it. I think that there are no such embedding, but I have no clue how to prove it.