In Fulton's book, page 34, it is stated that,
To compute the character of $V = Ind\space W$, note that $g \in G$ maps $\sigma W $ to $g \sigma W$, so the trace is calculated from those cosets $\sigma$ with $g \sigma = \sigma$, i.e., $s^{-1}gs \in H $ for $s \in \sigma$.
Can you clarify this for me? I don't see the connection to the representation matrix and traces.