I know that $F(x)=\dfrac{1}{1-x}$ is a generating function of $(1,1,1,1,\ldots)$ and $F(2x)$ is a generating function of $(1,2,4,8,16,\ldots).$
Then $G(x)=\dfrac{F(2x)-1}{x}=\dfrac{2}{1-2x}$ is a generating function of $(2,4,8,16,\ldots).$
A generating function of $(2,0,0,4,0,0,8,0,0,\ldots)$ is $G(x^3).$
A generating function of $(0,2,0,0,4,0,0,8,0,\ldots)$ is $xG(x^3).$
A generating function of $(0,0,2,0,0,4,0,0,8,\ldots)$ is $x^2G(x^3).$
A generating function of $(2,2,2,4,4,4,8,8,8,\ldots)$ is $$ H(x)=\left(1+x+x^2\right)G(x^3)=\left(1+x+x^2\right)\cdot \frac{2}{1-2x^3}. $$
Is this correct?
If the book gave the answer as $\frac{1+x+x^2}{1-x}$, that's wrong; your $H$ is correct. Indeed,
$$\begin{align*} \frac{1+x+x^2}{1-x}&=(1+x+x^2)(1+x+x^2+x^3+\cdots)\\ &=1+2x+3x^2+3x^3+\cdots \end{align*}$$