Let $G$ be a topological group with connected component $G^0$. Let $H$ be the subgroup of $G$ that consists of the elements that centralize $G^0$ and that act trivially on the component group $G/G^0$ by conjugation. It is clear that $$Z(G) Z(G^0) \subset H \,.$$ Does equality hold? If not, does it hold when $G/G^0$ is finite?
I asked myself this question while thinking about something vaguely related. I don't need it for any purpose, I'm just curious because this seems plausible but not easy to prove.