Derangements in wreath products of permutation groups

Given a finite group G acting on a set X let δ _k(G,X) denote the proportion of elements in G that have exactly k fixed points in X . Let 𝒮_n denote the symmetric group acting on [n]={1,2,… ,n} . For A⩽𝒮_m and B⩽𝒮_n , the permutational wreath product A≀ B has two natural actions and we give formulas for both, δ _k(A≀ B,[m]×[n]) and δ _k(A≀ B,[m]^[n]) . We prove that for k=0 the values of these proportions are dense in the intervals [δ _0(B,[n]),1] and [δ _0(A,[m]),1] . Among further results, we provide estimates for δ _0(G,[m]^[n]) for subgroups G⩽𝒮_m≀𝒮_n containing 𝒜_m^[n] .