Derangements in wreath products of permutation groups

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 $G\le {\mathrm{S}}_m\wr {\mathrm{S}}_n$ containing ${\mathrm{A}}_m^{[n]}$.