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?