Profinite groups with many elements with large nilpotentizer and generalizations

Given a profinite group $G$ and a family $\mathcal{F}$ of finite groups closed under taking subgroups, direct products and quotients, denote by $\mathcal{F}(G)$ the set of elements $g\in G$ such that $\{x\in G\mathrm{\mid }⟨g,x⟩isapro-\mathcal{F}group\}$ has positive Haar measure. We investigate the properties of $\mathcal{F}(G)$ for various choices of $\mathcal{F}$ and its influence on the structure of $G$.