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This paper is a fundamental study of the Real $2$-representation theory of$2$-groups. It also contains many new results in the ordinary (non-Real) case.Our framework relies on a $2$-equivariant Morita bicategory, where a novelconstruction of induction is introduced. We identify the Grothendieck ring ofReal $2$-representations as a Real variant of the Burnside ring of thefundamental group of the $2$-group and study the Real categorical charactertheory. This paper unifies two previous lines of inquiry, the approach to$2$-representation theory via Morita theory and Burnside rings, initiated bythe first author and Wendland, and the Real $2$-representation theory of$2$-groups, as studied by the second author.

Burnside rings for Real $2$-representation theory: The linear theory