  Given a finite group $G$ acting on a set $X$ let $\delta_k(G,X)$ denote the
proportion of elements in $G$ that have exactly $k$ fixed points in $X$. Let
$\mathrm{S}_n$ denote the symmetric group acting on $[n]=\{1,2,\dots,n\}$. For
$A\le\mathrm{S}_m$ and $B\le\mathrm{S}_n$, the permutational wreath product
$A\wr B$ has two natural actions and we give formulas for both, $\delta_k(A\wr
B,[m]{\times}[n])$ and $\delta_k(A\wr B,[m]^{[n]})$. We prove that for $k=0$
the values of these proportions are dense in the intervals
$[\delta_0(B,[n]),1]$ and $[\delta_0(A,[m]),1]$. Among further result, we
provide estimates for $\delta_0(G,[m]^{[n]})$ for subgroups $G\leq
\mathrm{S}_m\wr\mathrm{S}_n$ containing $\mathrm{A}_m^{[n]}$.
