**Question:** Find a finite group $G$, a subnormal subgroup $H$ of $G$, a real-valued irreducible $2$-Brauer character $\chi$ of $G$ and a real-valued irreducible $2$-Brauer character $\mu$ of $H$ such that $\mu$ occurs with odd multiplicity $\geq3$ in $\chi_H$.

**Context:** Let $N$ be a normal subgroup of a finite group $G$, let $p$ be a rational prime and let $\chi$ be an irreducible ($p$)-Brauer character of $G$. Then according to Clifford theory $\chi_N=e(\theta_1+\dots+\theta_t)$, where $e,t$ are positive integers and $\theta_1,\dots,\theta_t$ is a $G$-orbit on the irreducible Brauer characters of $N$. We say that $\chi$ lies over each $\theta_i$, and that $\theta_i$ has multiplicity $e$ in the restricted character $\chi_N$.

From now on $p=2$. Let $\theta$ be a real-valued irreducible Brauer character of $N$. Then by a recent result of Rod Gow and the author, there is a unique real-valued irreducible Brauer character $\chi$ of $G$ such that $\theta$ has odd multiplicity in $\chi_N$. Moreover this multiplicity is $1$ (so $\chi_N$ is the sum of the $G$-orbit of $\theta$). Equivalently, $\theta$ has a unique real-valued extension to its stabilizer in $G$.

For fans of solvable groups, this result follows from well-known properties of Isaac's canonical set B$_{2'}(G)\subseteq{\rm Irr}(G)$, and a result of I. M. Richards: if $|G:N|$ is odd then each real-valued ordinary irreducible character of $N$ has a unique real-valued extension to its stabilizer in $G$. Our proof, for all finite groups, uses a straightforward cohomological argument over a perfect field of characteristic $2$.

Suppose now that $H$ is a subnormal subgroup of $G$, and $\mu$ is real-valued irreducible Brauer character of $H$. Iterating the above result, there is a unique real-valued irreducible Brauer character $\chi$ of $G$ such that $\mu$ has odd multiplicity in $\chi_H$. Of course the irreducible constituents of $\chi_H$ may no longer be all $G$-conjugate. It is easy to see that the multiplicity must be $1$ if there exists $N$ with $H\unlhd N\unlhd G$.