  For a prime number $p$ and a free pro-$p$ group $G$ on a totally ordered
basis $X$, we consider closed normal subgroups $G^\Phi$ of $G$ which are
generated by $p$-powers of iterated commutators associated with Lyndon words in
the alphabet $X$. We express the profinite cohomology group $H^2(G/G^\Phi)$
combinatorically, in terms of the shuffle algebra on $X$. This partly extends
existing results for the lower $p$-central and $p$-Zassenhaus filtrations of
$G$.
