Equation in $GL(2,\mathbb{Z}_{2^{n-1}})$

Question

I'm now studying how to embed generalized quaternion group $Q_{2^n}=a^{2^{n-1}}=e$, $u^2=a^{2^{n-2}}$, $ua=a^{-1}u$ in $GL(k,\mathbb{Z}_{2^{n-1}}).$ I got the embedding in case $k=3$, but $k=2$ is still questionable. I checked for some small $n$'s this embedding in SAGE and found that $Q_{2^n}$ does not embedded in $GL(2,\mathbb{Z}_{2^{n-1}})$ in this case. This problem is equivalent to the problem of finding solutions of following matrix equation over $GL(2,\mathbb{Z}_{2^{n-1}}):$
$UA=A^{-1}U$, $ord$ $A=2^{n-1}$, $ord$ $U=4$, $U^2=A^{2^{n-2}}$.
Could someone help me to deal with this equation or maybe to suggest the other approach to the problem?