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## Induced character for non-injective homomorphisms

Any group homomorphism $\phi\colon H\to G$ gives rise to an induction/restriction adjunction between $G$-representations and $H$-representations: $$\hom_G(\phi_! M, N) \cong \hom_H(M, \phi^* N)$$ However, it seems that most textbooks and web pages about representation theory inexplicably consider only the case when $\phi$ is injective, i.e. exhibits $H$ as a subgroup of $G$. In this case, there are formulas for the character of $\phi_! M$ in terms of the character of $M$: $$\chi_{\phi_!(M)}(g) = \frac{1}{|H|} \sum_{k\in G \atop k^{-1} g k \in H} \chi_M(k^{-1} g k) = \sum_{\text{cosets } k H \atop k^{-1} g k \in H} \chi_M(k^{-1} g k) .$$ Can someone give a reference for versions of these formulas when $\phi$ is not injective?

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 Isn't it enough to replace $H$ bu $\phi (H)$ to reduce to the injective case ? – Paul Broussous Mar 1 2012 at 11:47

Exercise 7.1 in Serre's Linear Representations of Finite Groups gives a formula (without proof) in the case where $\phi$ is surjective. It is probably straightforward to compose this formula with your formula for the injective case to get the general formula.