Let A be a Dedekind domain of characteristic $p$ whose fraction field is $F$. Let $G$ be a finite group (not neccesarily abelian) whose order is not divisible by $p$.
Then the group algebra $F[G]$ is semisimple and the Wedderburn components are in correspondence with irreducible characters $\chi: G \rightarrow \overline{F}$. One finds that each simple component $A{\chi}$ is a matrix ring over the finite extension $F(\chi)$. This allows us to define a reduced norm $F[G]\rightarrow Z(F[G])$ by simply taking determinants in each component, and this can be extended to matrices over $F[G]$.
My question, although rather vague, is the following: given a subgroup $H$ of $G$, is there an obvious way to relate the reduced norm over $F[G]$ with the one over $F[H]$?
This question could be partially answered by giving a relationship between the Wedderburn components of each algebra. Say we have a Wedderburn component of $F[G]$ corresponding to a character $\chi$ of degree $n$: $M_n(F(\chi)$. If $\rho$ is an irreducible representation of $G$ whose character is $\chi$, we can find a decomposition of $\rho|_H$ in terms of irreducible representations of $H$, say a collection $\{\rho_j\}$ with characters $\psi_j$.
So each Wedderburn component $M_n(F(\chi))$ of $F[G]$, when restricting to $H$, will become a direct sum of Wedderburn components of $F[H]$ of the form $\bigoplus_j M_n(F(\psi_j))$. Is everything I've said up to now true? Can these components be treated canonically as a subring of $M_n(F(\chi))$?.
Thanks in advance