Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups $$ 0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0, $$ $$ 0\rightarrow G\rightarrow M_2\rightarrow H\rightarrow 0 $$ are considered equivalent is there is an isomorphism of compact Lie groups $M_1\rightarrow M_2$ making the rhomb-like diagram commute. Denote by $\mathrm{Ext}_{Lie}(G, H)$ the set of equivalence classes of short exact sequences of the above form.

We have a forgetful functor $F$ from the category of compact Lie groups to the category of groups. In its full image, one can also define the set of inequivalent extensions, $\mathrm{Ext}_{Grp}(\cdot, \cdot)$. Note that $F$ induces a well-defined map from $\mathrm{Ext}_{Lie}(G, H)$ to $\mathrm{Ext}_{Grp}(G, H)$. Is this map always an injection? What if we demand $G$ to be connected and $H$ to be zero-dimensional?

I do not think that this question can be answered by general nonsense, because analogous statement for the forgetful functor to closed manifolds does not hold (though I would be glad if I am proven wrong).