Level structures on $p$-divisible groups from the Morava $E$-theory of abelian groups

The close relationship between the scheme of level structures on the universal deformation of a formal group and the Morava $E$-cohomology of finite abelian groups has played an important role in the study of power operations for Morava $E$-theory. The goal of this paper is to explore the relationship between level structures on the $p$-divisible group given by the trivial extension of the universal deformation by a constant $p$-divisible group and the Morava $E$-cohomology of the iterated free loop space of the classifying space of a finite abelian group.