Let $G$, $H$ be topological groups and let $t \colon G \to H$ be a homomorphism, such that $t \colon G \to H$ is a topological crossed module. For a topological space $X$ we can define the nonabelian cohomology set $\check{H}^1(X, G \to H)$. There is a map of crossed modules from $1 \to H$ to $G \to H$ and this induces a map $$ \check{H}^1(X, H) \to \check{H}^1(X,G \to H) $$ What are the conditions for this map to be injective? If I express this problem in terms of classifying spaces, then I think I am asking for the fiber of the map $$ BH \to B(G \to H)\ . $$ In particular, my vague hope was that if $G$ is contractible, then the above map is injective. Is this true?
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The fiber of that map is well known to be $BG$, so what you expect is true. 


For results on the classifying space of (discrete) crossed modules, or more generally, crossed complexes, and in relation to homotopy classification and fibrations, see R. Brown, Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups, J. Homotopy and Related Structures 3 (2008) 331343. However the topological case has not been worked up in this format, as far as I know. 

