Here is a partial answer if $G$ is finite and Abelian, in this case we denote by $H^{\delta}$ the same group than $H$ with the discrete topology.The natural homomorphism $i:H^{\delta}\rightarrow H$ induces a continuous map $f:BH^{\delta}\rightarrow BH$.

In his paper entitled On the homology of Lie groups made discrete, Milnor conjectures the following fact: $f:BH^{\delta}\rightarrow BH$ induces an isomorphism of homology and cohomology with mod $p$ coefficients or more generally with any finite coefficient group.

Let $G$ be such finite coefficient group, the conjecture of Milnor says that $f$ induces an isomorphism $H^2(BH^{\delta},G)\rightarrow H^2(BH,G)$ and this is equivalent to saying that $Ext_{Lie}(G,H)$ is isomorphic to $Ext_{Gpr}G,H)$.

In that paper, Milnor shows its conjecture in different cases, for example if the identity component of the group $H$ is solvable. The paper of Milnor has been published a long time ago, perhaps there are other results in that direction.

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

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