Solution
= Solution
On the orbit $G/H$, an element $g\in G$ fixes the coset $xH$ exactly when
$$
gxH=xH
\quad\Longleftrightarrow\quad
x^{-1}gx\in H.
$$
The <character of a representation> of the corresponding <permutation representation> therefore is
$$
\boxed{
\chi_{\mathbb C(G/H)}(g)
=|(G/H)^g|
=\frac1{|H|}
|\{x\in G:x^{-1}gx\in H\}|
=(1_H\mathbin{\uparrow}^G)(g).}
$$
This is the <permutation character of a coset action>.