If we have $H$ a subgroup of $G$ and $U$ an $\mathbb{F}H$-module, $U\otimes_{\mathbb{F}} \mathbb{F}G$ is an $\mathbb{F}G$-module defined by $$(u\otimes \alpha)\cdot g=u\otimes (\alpha g)$$ for $u\in U, \alpha \in \mathbb{F}G$ and $g\in G$.
Moreover, $I_{U}=\langle u\otimes h\alpha - u\cdot h\otimes \alpha \mid u\in U, h\in H, \alpha \in \mathbb{F}G\rangle$ is an $\mathbb{F}G$-submodule. Finally, we define $$U\uparrow_{H}^{G}=(U\otimes \mathbb{F}G)/ I_{U}.$$
If $\phi$ is a character of $H$, I would like to understand the character afforded by $U\uparrow_{H}^{G}$.
I know that a basis of $U\uparrow_{H}^{G}$ is $\{b \otimes r + I_{U}\mid b\in B, r\in R\}$ where $B$ is and $\mathbb{F}$-basis of $U$ and $R=\{r_{1},\cdots,r_{n}\}$ is a set of representatives of the cosets of $H$ in $G$.
So in order to understand the induced character, first I need to know the induced representation (denote it by $\rho$) matrix.
To simplify the example, assume that $U= \mathbb{C}$. So it suffices to know the images of $1\otimes r_{i}+I_{\mathbb{C}}$ for $i\in \{1,\cdots,n\}$. How can I calculate the coefficients of that images with respect to the basis of $U\uparrow_{H}^{G}$?
Thanks in advance.