  Every word $w$ in the free group $F_r$ of rank $r$ induces a probability
measure (the $w$-measure) on every compact group $G$, by substitution of
Haar-random $G$-elements in the letters. This measure is determined by its
Fourier coefficients: the $w$-expectations $\mathbb{E}_w[\chi]$ of the
irreducible characters of $G$. For every compact group $G$, the wreath product
with the symmetric group $G\wr S_n$ has some natural irreducible characters
$\chi$, and we approximate $\mathbb{E}_w[\chi]$ for every word $w\in F_r$,
revealing new automorphism-invariant quantities of words that generalize the
primitivity rank $\pi(w)$. This generalizes previous works by
Parzanchevsky-Puder and Magee-Puder. We demonstrate applications to
automorphism groups of trees, investigate properties of the new invariants, and
show polynomial decay of $\mathbb{E}_w[\chi]$ also for wreath products with
more general actions.
