Let $\phi\colon H\to G$ be a group homomorphism, $V$ a representation of $H$, and $\chi$ the character of $V$. The induced character $\phi_!(\chi)$ of $f$ is the character of the induced $G$-representation
There is a formula for the induced character:
where the sum is over all pairs $(k\in G, h\in H)$ such that $k^{-1} g k = \phi(h)$.
This formula is usually given only in the case when $\phi$ is injective, when it can be re-expressed as a sum over cosets. The case when $\phi$ is surjective is Exercise 7.1 of (Serre) and the general case is easy to put together from these. It can also be derived abstractly using bicategorical trace.
J-P. Serre, Linear Representations of Finite Groups
MO question about the non-injective case.