  Every word $w$ in $F_r$, the free group of rank $r$, induces a probability
measure (the $w$-measure) on every finite group $G$, by substitution of random
$G$-elements in the letters. This measure is determined by its Fourier
coefficients: the $w$-expectations $E_w[\chi]$ of the irreducible characters of
$G$. For every finite group $G$, every stable character $\chi$ of $G\wr S_n$
(trace of a finitely generated $FI_G$-module), and every word $w\in F_r$, we
approximate $E_w[\chi]$ up to an error term of $O(n^{-\pi(w)})$, where $\pi(w)$
is the primitivity rank of $w$. This generalizes previous works by Puder,
Hanany, Magee and the author. As an application we show that random Schreier
graphs of representation-stable actions of $G\wr S_n$ are close-to-optimal
expanders. The paper reveals a surprising relation between stable
representation theory of wreath products and not-necessarily connected
Stallings core graphs.
