Real Representations of $C_2$-Graded Groups: The Linear and Hermitian Theories

We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the category hermitian representations, as an $\infty$-category, is equivalent to a category of usual representations.
Comment: Version 2: final published version, to appear in Higher Structures