Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:

$$
1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1
$$

There exists a forgetful functor $g:Ext_{Lie}(G,H)\rightarrow Ext_{Gpr}(G,H)$ obtained by using the forgetful functor $Lie\rightarrow Gpr$ from the category of Lie groups to the category of groups.

Recently https://mathoverflow.net/questions/324870/extensions-of-compact-lie-groups/324900#324900 , the following question has been asked: is $g$ injective ?

Let $H$ be a Lie group and $H^{\delta}$  the underlying group of $H$ endowed with the discrete topology, the canonical embedding $i:H^{\delta}\rightarrow H$ induces a morphism $f:BH^{\delta}\rightarrow BH$. In his paper entitled the homology of Lie groups make discrete, Milnor has conjectured that  $f$ induces an isomorphism between the homology and cohomology with finite coefficients of $BH^{\delta}$ and $BH$. 

This enables to give a partial answer to the previous question if $G$ is finite and commuative.

Question: Is the conjecture of Milnor has been already proved ? Or in what cases it is known to be true ?

Milnor, J. On the homology of Lie groups made discrete.

Comment. Math. Helvetici 58 (1983) 72-85.