I have a new relevant question. Let U, V, W be connected Lie groups, so that U is the semidirect product of V and W. By Schreier's theorem (Pontrjagin, Theorem 61), each group has a unique (up to an isomorphism) universal cover.
Question. If $U = V \rtimes W$, then is there a connection between $\tilde U, \tilde V, \tilde W$? If so, what would be the proof? Physicists assume that
$$\tilde U \simeq \tilde V \rtimes \tilde W$$ (see the case of Lorentz and Poincaré groups), so I am thinking & hoping a mathematician should know better.