Let $G=S_3$.
\begin{array}{|c|c|c|} \hline & e & (123) & (12) \\ \hline \chi_0 & 1 &1 & 1\\ \hline \chi_1 & 1 & 1 & -1 \\ \hline \chi_2 & 2 & -1 & 0 \\ \hline \end{array}
Let $H=C_3$. \begin{array}{|c|c|c|} \hline & e & (123) & (12) \\ \hline \chi_0 & 1 &1 & 1\\ \hline \chi_1 & 1 & \zeta & \zeta^2 \\ \hline \chi_2 & 1 & \zeta^2 & \zeta \\ \hline \end{array}
Then if $\chi=(a,b,c)$ is a representation of $G=S_3$, how does $Res_H\chi=(a,b,b)$?
Also I am not sure why (12) is a conjugacy class for $C_3$.
I know the definition $(Res_H\chi)(h)=\chi(h) \ \ \ \ \forall h \in H$.