We determine those pseudovarieties of groups ${\bf H}$ for which the power monoids $P(G)$, ranging over all groups $G$ in ${\bf H}$, satisfy the same profinite identities (i.e. pseudoidentities) as all semidirect products of ${\cal J}$-trivial monoids by groups in ${\bf H}$. That is, in the language of finite monoid theory, we characterize all solutions to the pseudovariety equation ${\bf PH}\,{=}\,{\bf J}\ast {\bf H}$. The characterization is in terms of the geometry of the Cayley graphs of the free pro-${\bf H}$ groups as well as in terms of the pro-${\bf H}$ topology of a finitely generated free group.