Pro-$\mathbf H_p$ density in the free group of rank two: a Frobenian criterion on the mod-$p$ torus

Let $F$ be the free group of rank two and, for a prime $p$, let $\mathbf H_p=\mathbf G_p*\mathbf{Ab}_{p-1}$ be the pseudovariety of finite groups having a normal $p$-subgroup with abelian quotient of exponent dividing $p-1$. For $H$ of rank two with $\ab_F(H)=F^{\ab}$ we determine the set $\mathfrak