Find a Möbius Transformation on complex plane, that moves $|z|<1$ to $|w|>2$ and point (0, 0) to (4, 0).
If we don't concern second condition, we can use $w(z) = \frac{2}{z}$. But later we can't apply the transition.
So I decided to transform cirle to a half-plane, then mirror it, and then convert it back to the cirle but with another radius. But this approach is very hard, and I'm stuck finding correct constants.
$w(z) = 2/z$ maps the unit disk $\Bbb D$ to $|w| > 2$ with $w(1/2) = 4$. So we need to pre-compose this with a Möbius transformation $T$ which maps the unit disk onto itself and $T(0) = 1/2$.
The conformal automorphisms of the unit disk are well-known, that are exactly the Möbius transformations of the form $$ T(z) = e^{i \lambda} \frac{z-a}{1-\overline a z } $$ with $a \in \Bbb D$ and $\lambda \in \Bbb R$.
Choosing (for example) $\lambda = 0$ and $a = -1/2$ gives $T(0) = 1/2$, so that the desired mapping is $$ z \mapsto w(T(z)) = 2 \frac{1+z/2}{z+1/2} \, . $$