Let $\gamma$ be the unit circle. Find a Möbius transformation that transforms $\gamma$ onto $\gamma$ and transforms 0 to $\frac{1}{2}$.

Question

Let $\gamma$ be the unit circle. Find a Möbius transformation that transforms $\gamma$ onto $\gamma$ and transforms 0 to $\frac{1}{2}$.

I think it's quite easy to find a Möbius transformation that transforms $\gamma$ onto $\gamma$, since Möbius transformation always maps circle to circle or line. We can actually choose three points on $\gamma$ than solve the Möbius transformation by pluging in numbers. But when we also need $0 \rightarrow \frac{1}{2}$, I have a problems find such map.

And to check the “onto” fashion, can we simply say that the inverse is also a Möbius transformation?

Thanks for the help!