Let $\widehat F(k)$ be the free profinite group on $k$ generators and let $p$ be a prime. Then there is a canonical projection $\pi\colon \widehat F(k)\to \widehat F_p(k)$ where $\widehat F_p(k)$ is the free pro-$p$ group on $k$-generators. It is well known that a free pro-$p$ group is projective in the category of profinite groups and so $\pi$ splits via a continuous homomorphism $\psi\colon \widehat F_p(k)\to \widehat F(k)$.
I would like to know if anybody knows an explicit splitting. To be more precise, let $X_k =\{x_1,\ldots, x_k\}$ be a free generating set for $\widehat F(k)$. I would like explicit sequences $\{w_{i,n}\}$, for $i=1,\ldots, k$, of words over $X_k$ such that $w_{i,n}\to w_i$ with $\pi(w_i) = \pi(x_i)$ and $\overline {\langle w_1,\ldots, w_k\rangle}$ a free pro-$p$ group.
For example, if $k=1$, then I know how to write down a sequence converging to an element giving a splitting of $\pi\colon \widehat {\mathbb Z}\to \mathbb Z_p$. Namely, if $x_1$ is the free generator of $\widehat{\mathbb Z}$ and if $p_1,p_2,\ldots$ is a list of the primes other than $p$. Then $x_1^{(p_1p_2\cdots p_n)^{n!}}$ converges to an element generating a copy of $\mathbb Z_p$ and mapping to $\pi(x_1)$.