What space classifies bundles of K(pi,1)'s? Given a discrete group $G$ (not assumed to be abelian), is there a nice construction of a topological space which classifies bundles of $K(G,1)$'s?  I guess I should take something like $B\operatorname{Aut}(K(\pi,1))$ where $\operatorname{Aut}(K(\pi,1))$ is the monoid of self homotopy equivalences of some fixed $K(\pi,1)$.  I don't know if this makes any sense, though.
Morally speaking, things are defined as follows.  A bundle of $K(\pi,1)$'s over a space $X$ is a fibration $Y\to X$ whose fibers are $K(\pi,1)$'s.  Two bundles of $K(\pi,1)$'s $Y_1,Y_2\to X$ are said to be isomorphic iff there is a map $Y_1\to Y_2$ of spaces over $X$ which is a homotopy equivalence on each fiber.
 A: For any space $F$ you can form the topological monoid $Aut(F)$ and take its classifying space. That will classify Hurewicz fibrations with fiber $F$. A little care is needed since $Aut(F)$ won't generally have the homotopy type of a CW complex unless $F$ is a finite CW complex, which $K(G,1)$ is usually not. For the proof it is sensible to take base spaces of the homotopy types of CW complexes.  One source for a complete proof is "Classifying spaces and fibrations", #15 on my web page.  (There are two obvious notions of the right equivalence relation on fibrations, but these give the same answer, as shown in "Fiberwise localization and completion'', #31 on my web page.)
A: To put another context to the above comments and answers, recall that a crossed module consists of a morphism $\mu: M \to P$  together with an action of $P$ on the right of the group $M$ satisfying the two axioms 


*

*$\mu(m^p)= p^{-1}\mu(m) p$; 

*$ m^{-1} nm = m^{\mu n}$,
for all $m,n \in M, p \in P$. This definition is due to J.H.C. Whitehead in 1946. 
Examples of crossed modules are:


*

*the inner automorphism crossed module $\chi: M \to Aut(M)$ for any group $M$; 

*the inclusion $M \to P$ of a normal subgroup $M$ of $P$; 

*the zero map $0: M \to P$ for any right $P$-module $M$;

*the induced map $\pi_1(F) \to \pi_1(E)$ for any pointed fibration $F \to E \to B$; 
and others!
Any such crossed module has a classifying space $B(M \to P)$ whose homotopy groups are trivial above dimension $2$, and with $\pi_1 \cong Coker \mu$, $\pi_2 \cong Ker \mu$. More details of these facts are in the book EMS Tract 15. Crossed modules are equivalent to group objects in the category of groupoids, and this gives one way of defining the classifying space, using bisimplicial sets, and are conveniently regarded as $2$-dimensional versions of groups, since they model pointed weak homotopy $2$-types. Note that the second homotopy group, even considered as a module over $\pi_1$, is generally  but a pale shadow of the homotopy $2$-type. 
There is also a Seifert-van Kampen type theorem with values in crossed modules, and this allows some computations of nonabelian second relative homotopy groups. See again the EMS Tract 15. 
