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Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting, as well as $F_1, \ldots, F_k$. If they are similar, i.e., there exists $T \in GL_n(\mathbb{C})$ such that $$G_i = T^{-1} F_i T$$ for all $i \in [k]$, does there exist $V \in U_n$, where $U_n$ is the group of unitary matrices, such that $$G_i = V^{-1} F_i V$$ for all $i \in [k]$.

Generally, let $\pi, \sigma : G \to U_n(\mathbb{C})$ be two unitary representations of finite group. If $\pi, \sigma$ are equivalent, i.e., there exists $T \in GL_n(\mathbb{C})$ such that $T \pi(g) = \sigma(g) T$ for all $g \in G$, is $\pi$ and $\sigma$ unitarily equivalent, that is, there exists $T' \in U_n$ such that $T' \pi = \sigma T'$?
Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting, as well as $F_1, \ldots, F_k$. If they are similar, i.e., there exists $T \in GL_n(\mathbb{C})$ such that $$G_i = T^{-1} F_i T$$ for all $i \in [k]$, does there exist $V \in U_n$, where $U_n$ is the group of unitary matrices, such that $$G_i = V^{-1} F_i V$$ for all $i \in [k]$.