Topological Aspects in Representation Theory of 2-Graded 2-Groups

A 2-graded group is a pair: a group G and its index two subgroup H (even elements). A 2-graded 2-group is a 2-group GG such that the fundamental group $\pi_1(GG)$ is 2-graded.
They have several representation theories: 4 for groups, at least 8 for 2-groups. The interplay between them have interesting topological aspects that I will discuss in my talk.

The talk is based on the same recent works with James Taylor (Oxford) and Matthew B. Young (Utah State), as my November talk at the Representation Theory Oberseminar. 
However, it will be focused more on the topological aspects of the work. I will structure the talk in a way that caters for both those who attended the earlier talk and those who did not.