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?