We prove Clifford theoretic results on the representations of finite groups which only hold in characteristic $2$. Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $\varphi$ be an irreducible $2$-Brauer character of $N$ which is self-dual. We prove that there is a unique self-dual irreducible Brauer character $\theta$ of $G$ such that $\varphi$ occurs with odd multiplicity in the restriction of $\theta$ to $N$. Moreover this multiplicity is $1$. Conversely if $\theta$ is an irreducible $2$-Brauer character of $G$ which is self-dual but not of quadratic type, the restriction of $\theta$ to $N$ is a sum of distinct self-dual irreducible Brauer character of $N$, none of which have quadratic type. Let $b$ be a real $2$-block of $N$. We show that there is a unique real $2$-block of $G$ covering $b$ which is weakly regular.

中文翻译：

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特征二和正态子群中的自对偶模

我们证明了 Clifford 理论结果对仅在特征 $2$ 中成立的有限群的表示。令$G$为有限群，令$N$为$G$的正规子群，令$\varphi$为$N$的不可约$2$-Brauer字符，自对偶。我们证明了 $G$ 存在唯一的自对偶不可约 Brauer 字符 $\theta$，使得 $\varphi$ 在 $\theta$ 到 $N$ 的限制中以奇重数出现。此外，这种多重性是 1 美元。相反，如果 $\theta$ 是 $G$ 的不可约 $2$-Brauer 字符，它是自对偶但不是二次型，则 $\theta$ 对 $N$ 的限制是不同的自对偶不可约 Brauer 的和$N$ 的字符，没有一个是二次型的。让 $b$ 成为一个真正的 $2$-$N$ 块。我们表明存在一个独特的真实 $2$-$G$ 块覆盖 $b$，这是弱正则的。