Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H \leq_c^\infty F$. Suppose that there exists a discrete $H$-module structure on $M = (\mathbb{Z}/p\mathbb{Z})^m$ for some $m \in \mathbb{N}$ such that $H^1(H,M)$ is finite.
Must $L/N$ be free pro-$p$?
The case when one can take $M = \mathbb{Z}/p\mathbb{Z}$ and trivial as an $H$-module (every element of $H$ acts as an identity) and $H^1(H,M) = \{0\}$ is already of interest.
In this special case, I want to show that $L/N$ is free pro-$p$ given that $\text{Hom}(H,\mathbb{Z}/p\mathbb{Z}) = \{0\}$.
