  This article is dedicated to the asymptotic geometry of wreath products $F\wr
H := \left( \bigoplus_H F \right) \rtimes H$ where $F$ is a finite group and
$H$ a one-ended finitely presented group. Our main result is a complete
classification of these groups up to quasi-isometry. More precisely, given two
finite groups $F_1,F_2$ and two finitely presented one-ended groups $H_1,H_2$,
we show that $F_1 \wr H_1$ and $F_2 \wr H_2$ are quasi-isometric if and only if
either (i) $H_1,H_2$ are non-amenable quasi-isometric groups and $|F_1|,|F_2|$
have the same prime divisors, or (ii) $H_1,H_2$ are amenable, $|F_1|=k^{n_1}$
and $|F_2|=k^{n_2}$ for some $k,n_1,n_2 \geq 1$, and there exists a
quasi-$(n_2/n_1)$-to-one quasi-isometry $H_1 \to H_2$. The article also
contains algebraic information on groups quasi-isometric to such wreath
products. This can be seen as far reaching extension of a celebrated work of
Eskin-Fisher-Whyte who treated the case of $H=\mathbb{Z}$. Our approach is
however fundamentally different, as it crucially exploits the assumption that
$H$ is one-ended. Our central tool is a new geometric interpretation of
lamplighter groups involving natural families of quasi-median spaces.
