### A note on the continuous extensions of injective morphisms between free groups to relatively fre profinite groups.

Let be a pseudovariety of finite groups such that free groups are residually , and let φ: F(A) → F(B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension φ' of φ between the pro- completions of F(A) and F(B) is also injective. In particular, if is extension-closed, this is the case if and only if φ(F(A)) and its pro- closure in F(B) have the same rank. We examine a number of situations where the injectivity of φ' can be...