The character of the regular representation $\chi_R$ satisfies
$$\chi_R(g)=\begin{cases} |G| & g=id \\ 0 & \textrm{otherwise}\end{cases},$$
(https://en.wikipedia.org/wiki/Regular_representation). In a way, one could say that the character of the regular representation projects the group to the identity subgroup. In this line, is there a generalization where one projects to a generic subgroup? More precisely, exists some representation $\rho_K$ such that
$$\chi_K(g)=\begin{cases} 1 & g\in K \\ 0 & \textrm{otherwise}\end{cases}$$
for a given subgroup $K\subset G$?
This is not a character if $K$ is a proper subgroup of $G$. For suppose $\chi$ were a character: take the Frobenius inproduct with the principal character $1_{G}$, then the multiplicity is $\frac{|K|}{|G|}$. This must be a non-negative integer, and this only occurs when $K=G$.
Yet another approach, if it would be a character then it is linear and cannot have values equal to $0$, being a homomorphism to $\mathbb{C}^*$.
The right generalization of the regular character would be the induced character $1_{K}^G$. This is the permutation character of $G$ acting on the right cosets of $K$ by multiplication from the right. The regular character of $G$ is $1_{1}^G$, so the supertrivial character induced to the whole group.