 Solution

Write an element of $H\wr \mathbb{Z}$ as $(f,t^n)$. If it is central, commuting with $t$ makes the finitely supported lamp configuration $f$ invariant under translation. The only such configuration on the infinite set $\mathbb{Z}$ is the identity, so $f=1$. If $n\ne 0$, then $t^n$ moves a nonidentity lamp at position zero to position $n$ and does not commute with it. Therefore $n=0$, and