Let $X$ be a random variable with generating function $G_X(s) = \frac{1}{2}(1+s)/(2-s)$. We toss a fair coin, then I want to find the generating function of a random variable $Z$ which is defined as $Z = X$ if the coin is head, and $Z = 2X$ if the coin is tail.
Could you help to see if my solution is correct? I do not think it is easy to simplify my final result, so I am not sure if it is correct or not.
Your sollution is overly complicated. The easy path to finding the probability generating function of the product of independent random variables is to use the definition.
$$\mathsf G_X(t)=\mathsf E(t^X)$$
Let $Y$ be the random variable resulting in $1$ if the coin toss is head and $2$ if it is tails. So $Z=YX$ and $Y\sim\mathcal U\{1,2\}$.
$$\begin{align}\mathsf G_Z(s)&=\mathsf E(s^Z)\\&=\mathsf E(s^{YX})\\&=\mathsf E(\mathsf E(s^{YX}\mid Y))&&\text{Law of Total Expectation}\\&=\mathsf E(\mathsf E(s^{YX}))&&\text{Independence}\\&=\mathsf E(\mathsf E({(s^Y)}^X))\\&~~\vdots\end{align}$$