 Solution

Let $V$ afford the given group representation of $H$. Its induced representation is the left $G$-moduleChoose representatives $t$ of the left cosets $G/H$; then it is the direct sum of the spaces $t\otimes V$. If $gt=t^{\prime }h$ with $h\in H$, the action sends $t\otimes v$ to $t^{\prime }\otimes \Psi (h)v$. This is independent of the representatives up to equivalence, and its dimension is $[G:H]\dim V$.

Only cosets fixed by $g$ contribute to its trace. Their contribution is $\psi (t^{-1}gt)$, giving the induced character formulaFor an arbitrary class function $f$ on $H$, the same averaging formula defines its induced class function on $G$. Equivalently, extend $f$ by zero outside $H$ and average $|H{|}^{-1}{\sum }_{x}f(x^{-1}gx)$. This definition is linear and does not require $f$ to be an actual character.

For a complex character $\psi$ of a finite group $G$ afforded by a representation $\rho$, its determinant character isMultiplicativity of the determinant makes this a linear character; it does not mean the determinant of the scalar value $\psi (g)$. The definition depends only on the character because characters determine the equivalence class of a finite-dimensional complex representation. In particular the determinant character is trivial on the derived subgroup $G^{\prime }$.