I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie" and I was wondering if I could ask for help with understanding one part of the argument. He says that if you have a connected Lie group $G$ and a connected abelian Lie group $H$, and you assume that $G$ is not perfect and $H$ is not simply connected, then that means that the quotient of the group $\mathrm{Hom}(G,H)$ by the subgroup consisting of those abstract homomorphisms which lift to an abstract homomorphism of the universal covers is equal to a product of $2^{\aleph_{0}}$ copies of $\hat{\mathbb{Z}}/\mathbb{Z}$. I am not sure why this is true and indeed one problem I have is that I am not even sure what group structure is intended on $\mathrm{Hom}(G,H)$ (edit: this is now clear, I was not seeing it before because I forgot $H$ was abelian). I was wondering if anyone could give me a few hints with working it out.

Perhaps it would help to mention that in the context he discusses the fact that if you let $\tilde{G}$ be the universal cover of $G$ then its quotient by its derived group is a vector space over $\mathbb{R}$, and he also introduces a basis for it as a vector space over $\mathbb{Q}$, and then he talks about how you can use this to describe the homomorphisms from $\tilde{G}$ into $\tilde{H}$ (remembering that we are assuming $H$ is abelian). I get that bit. But I am not sure how you would describe the homomorphisms from $\tilde{G}$ into $H$ in terms of the ones that lift to a homomorphism from $\tilde{G}$ into $\tilde{H}$.