We know that if $\{ e\}<G$ is the trivial subgroup and $\chi_0$ is the (necessarily) trivial character of $\{e\}$, then the induced character in $G$ can be written neatly as
$$
\textrm{Ind}_{\{ e \}}^{\ G}(\chi_0) = \sum_{\chi}\chi(1)\chi,
$$
where the sum is taken over the irreducible characters of $G$.
But if $H < G$ is any subgroup and $\chi_0$ is the trivial character of $H$, can we find a similarly neat expression for
$$
\textrm{Ind}_{H}^{G}(\chi_0)?
$$
Best Answer
If $H$ is a subgroup of $G$, then inducing the trivial character $(1_H)^G$ can be interpreted as a transitive permutation character: here $G$ acts by right multiplcation on the right cosets of $H$ in $G$. There is a lot to say about this character and in general it does not split in irreducibles as neatly as the regular character $(1_{1})^G$ you mentioned. There are many boooks in which you can find properties of the permutation character, I.M. Isaacs, Character Theory of Finite Groups, Chapter 5, being one of them. Another one here.