  We use the structure of finite-dimensional graded algebras to develop the
theory of antilinear representations of finite $C_2$-graded groups. A finite
$C_2$-graded group is a finite group with a subgroup of index 2. In this theory
the subgroup acts linearly, while the other coset acts antilinearly. We
introduce antilinear blocks, whose structure is a crucial component of the
theory. Among other things, we study characters and Frobenius-Schur indicators.
As an example, we describe the antilinear representations of the $C_2$-graded
group $A_n \leq S_n$.
