Let G be a group and $H\le G$ be a subgroup. Let $W$ be a representation of H for which the associated character is $\eta$ and suppose moreover that $\{\chi_1,\ldots,\chi_r\}$ are the irreducible characters of $G$ for which the respective representations are $V_1,\ldots,V_r$.
Now $\operatorname{Ind}^G_HW$ is a representation of G for which we will denote the character $\chi$. Using Frobenius reciprocity, I can often express $\operatorname{Ind}^G_H\eta=\sum\limits_{k=1}^r a_k \chi_k$ but then it is not clear why $\operatorname{Ind}^G_HW=\bigoplus_{k=1}^r V_k$ (or maybe it is another direct sum, I am not sure).