Let $G$ be a finite group and $H$ a normal subgroup. Let $F$ be a local field of characteristic zero. The group algebras $F[H]\subset F[G]$ are semisimple by Maschke’s theorem. Consider their Wedderburn decompositions. A Wedderburn component $M_{n_\chi}(D_\chi)$ of $F[G]$ corresponds to an irreducible characters $\chi$ of $G$, and similarly a Wedderburn component $M_{n_\eta}(D_\eta)$ of $F[H]$ corresponds to irreducible characters $\eta$ of $H$, where $D_\chi$ resp. $D_\eta$ are skew fields.
If $\eta$ is an irreducible constituent of the restriction $\chi_H$, is there a relationship between the Hasse invariants of $D_\chi$ and $D_\eta$?
Since the question involves a normal subgroup of a finite group, it sounds to me like the answer should involve Clifford theory, but I failed to find references for this.