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

Let $\mathbf{V}$ be a pseudovariety of finite groups such that free groups are residually $\mathbf{V}$, and let $\varphi\colon F(A)\rightarrow F(B)$ be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension $\hat\varphi$ of $\varphi$ between the pro-$\mathbf{V}$ completions of $F(A)$ and $F(B)$ is also injective. In particular, if $\mathbf{V}$ is extension-closed, this is the case if and only if $\varphi(F(A))$ and its pro-$\mathbf{V}$ closure in $F(B)$ have the same rank. We examine a number of situations where the injectivity of $\hat\varphi$ can be asserted, or at least decided, and we draw a few corollaries.