A projective profinite group whose smallest embedding cover is not projective (Q1320024)

A profinite group \(G\) has the embedding property if for each pair of epimorphisms (\(\varphi: G \to A\), \(\alpha : B \to A\)) where \(B\) is a finite quotient of \(G\) there exists an epimorphism \(\gamma : G \to B\) such that \(\alpha \circ \gamma = \varphi\). An epimorphism \(\pi : E \to H\) of profinite groups such that \(E\) has the embedding property is an embedding cover of \(H\) if in addition, for each embedding cover \(\varphi : G \to H\) there exists an epimorphism \(\theta: G \to E\) such that \(\pi \circ \theta = \varphi\). If a profinite group \(G\) has the embedding property, then so does its smallest projective cover \(\widetilde G\). The goal of the paper is to give a negative answer to the following problem 23.16 from the book ``Field Arithmetic'' (1986; Zbl 0625.12001) by \textit{M. D. Fried} and \textit{M. Jarden}. Problem. Let \(H\) be a finitely generated profinite group. a) Is \(E(H)\) projective whenever \(H\) is? b) Is \(E(\widetilde H)\) isomorphic to the smallest projective cover of \(E(H)\)? A counterexample to both parts of the problem is constructed in the paper.