2-categories of symmetric bimodules and their 2-representations

In this article we analyze the structure of 2-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting 2-category is fiat (in the sense of Mazorchuk and Miemietz (2011)) and classify simple transitive 2-representations of this 2-category (under some mild technical assumption). We also study several classes of examples in detail.