Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence $$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$ where $G_c$ is the normal subgroup which contains all elements in the same connected component as the identity element, and $G/G_c$ can be thought of as the "finite part" of $G$. Suppose A is a finite $G/G_c$ module (as well as a $G$ module).

The question is: Is the cohomology map $$H^3(G/G_c, A)\rightarrow H^3(G, A)$$ induced by the projection $p:G\rightarrow G/G_c$ always injective?

The background is as follow: given any finite group T, a homomorphism $G\rightarrow T$ always factors through $G/G_c$. I wish to prove (or disprove) the similar statement for $\mathcal{T}$ a "finite" 2-group.