I am reading some notes that concerns permutation modules, but inside there is something I don't understand.
Suppose $G$ acts on a set $X$ and denote $\mathbb{C}X$ the permutation module. It is said that given $x\in X$, $\operatorname{Orb}(x)$ is isomorphic as $G$-set to $G/H$ for some subgroup $H\le G$ and so $\mathbb{C}[G/H]$ is a direct summand of $\mathbb{C}X$, with character $\operatorname{Ind}^G_H1$ ($\operatorname{Ind}$ is the induction).
What exactly is the action on $G/H$ for the isomorphism? What is the subgroup $H$ and why? Why does $\mathbb{C}[G/H]$ have character $\operatorname{Ind}^G_H$?
Edit
If this is the action, then taking $H=G_x$ (the stabilizer of $x$) then we have $orb(x)≅G/G_x$ (isomorphism of $G$-set). Then I know the orbits partition $X$, i.e. $X=orb(x_1)⊔…⊔orb(x_r)$ so we have $\mathbb{C}X=Corb(x_1)⊕…⊕Corb(x_r)≅C[G/Gx_1]⊕…⊕C[G/Gx_r]$ but then it seems to me that $Ind^G_H$ is a character of $G$. Why do they claim it is a character of $C[G/H]$?
The action of $G$ on $G/H$ is left multiplication (assuming the action of $G$ on $X$ is a left action). From here you should be able to work out the rest of your questions.