Let $G$ be a connected topological group and let $p:\tilde{G}\to G$ be a universal covering of $G$. Then $\tilde{G}$ is also a topological group and $p$ is a continuous homomorphism.
My question is: How do we define multiplication in $\tilde{G}$? We can define a map $$p\times p:\tilde{G}\times \tilde{G} \to G, \quad (x,x)\mapsto (p(x),p(x))$$
and try to lift it to a map $\tilde{G} \times \tilde{G} \to \tilde{G}$. Such a lift exists iff $(p\times p)_{*} \pi_1(\tilde{G}\times \tilde{G}) \subset p_*\pi_1(H)$.
Is this condition satisfied? If so, how do we verify that this actually gives us a group structure?