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?



Induced character for non-injective homomorphisms
[Exercise 7.1 in Serre's *Linear Representations of Finite Groups*](http://books.google.com/books?id=NCfZgr54TJ4C&pg=PA57&lpg=PA57) 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.


