Restriction of real irreducible 2-Brauer characters to subnormal subgroups

Question

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 $H\unlhd N\unlhd G$.

Answer 1

Jürgen Müller has provided me with two families of examples, where the restriction multiplicities are arbitrarily large and odd. We describe these below, with his kind permission.

First let $p$ be a prime such that $p\equiv1$ (mod $4$), let $L :=\operatorname{GL}_2(p)$, let $S$ be a Sylow $2$-subgroup of $L$, and set $N := N_L(S)$. Then $S =C_s\wr C_2$, where $s$ is the largest $2$-power dividing $p-1$ and the outer $2$ can be chosen as the involution $t:=\begin{bmatrix}0&1\\1&0\end{bmatrix}$. Moreover, $N = \operatorname{Z}(L) S = C_{(p-1)/s}\times S$ has order $2s(p-1)$.

Next, let $E = Z_p + Z_p$ be the natural $L$-module, and let $G := E\rtimes N$ be the natural semidirect product. Now let $u \in E$ be an eigenvector of $t$ w.r.t. the eigenvalue $-1$. Then $H := \langle u,t\rangle$ is dihedral of order $2p$. Further, $H$ is subnormal in $G$ as it is normal in $E\rtimes(\operatorname{Z}(L)\times\langle t\rangle)$ and $G/(E\rtimes\operatorname{Z}(L))$ is nilpotent.

Now all irreducible $2$-modular Brauer characters of $H$ are self-dual, and, next to the trivial character $1$, there are $(p-1)/2$ characters $\phi_i$ of degree $2$. Finally, $G$ has a unique, hence self-dual, irreducible $2$-modular Brauer character $\phi$ of degree $s(p-1)$, whose restriction to $H$ decomposes as $(p-1)$ copies of $1_H$ and $(s-1)$ copies of $\sum_i \phi_i$. Here $s-1 \geq 3$ is odd, and can be made arbitrarily large by chosing $p$ appropriately.

The case $p\equiv -1$ (mod $4$) can be treated along very similar lines, only the description of the Sylow $2$-subgroup $S$ of $L :=\operatorname{GL}_2(p)$ and the Clifford theory became slightly more tricky. Letting $s$ be the largest $2$-power dividing $p+1$, we have $S = \operatorname{QD}_{4s}$, a quasi-dihedral group, and $\operatorname{N}_L(S) = C_{(p-1)/2}\times S$.

Choosing $G$ and $H = D_{2p}$ as before, utilising a suitable non-central involution of $S$, again there is a (unique) self-dual, irreducible $2$-modular Brauer character $\phi$ of $G$ of degree $s(p-1)$, whose restriction to $H$ decomposes as $(p-1)$ copies of $1_H$ and $(s-1)$ copies of $\sum_i \phi_i$.

For the whole series, the Brauer character $\phi$ has defect $1$, hence belongs to a block of defect $1$ and is the only irreducible Brauer character of this block.