Question: How should we interpret and understand the classifying space $B^nG$? Is that Eilenberg-MacLane space $K(G,n)$?
What one can learn about $BG$ follows the basic: A classifying space $BG$ of a topological group $G$ is the quotient of a weakly contractible space $EG$ (i.e. a topological space for which all its homotopy groups are trivial) by a free action of $G$. It has the property that any $G$ principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle $EG \to BG$. For a discrete group $G$, $BG$ is, roughly speaking, a path-connected topological space $X$ such that the fundamental group of $X$ is isomorphic to $G$ and the higher homotopy groups of $X$ are trivial, that is, $BG$ is an Eilenberg-MacLane space, or a $K(G,1)$.
So if $G$ is a finite discrete group, what are the conditions of the higher homotopy groups of $B^nG$?