Proposition 2.3

For every profinite group $J$ in universe $u$, every finite group $N$ in universe $v$ with the discrete topology, and every jointly continuous action of $J$ on $N$ by group automorphisms, let $p$ be a natural prime and assume that every $n\in N$ satisfies $n^{p^k}=1$ for some $k\in\mathbb N$. Suppose that $N\rtimes J$, with multiplication $(n,j)(n',j')=(n(j\cdot n'),jj')$ and the product topology, is prosupersolvable. Let $Q\le J$ be closed, and assume that for every open normal subgroup $U$ of $J$, writing $\overline Q$ for the image of $Q$ under $J\to J/U$, every prime dividing $|\overline Q|$ is at most $p$, and every prime dividing the index $[J/U:\overline Q]$ is greater than $p$. For $A\le J$, write $H^1(A,N)$ for the set of continuous maps $f:A\to N$ satisfying $f(xy)=f(x)(x\cdot f(y))$, modulo the equivalence relation $f\sim g$ if there is one $n\in N$ such that $g(x)=n^{-1}f(x)(x\cdot n)$ for every $x\in A$; its distinguished element is the class of the constant map $1$. Then the restriction map $H^1(J,N)\to H^1(Q,N)$, $[f]\mapsto[f|_Q]$, is both injective and surjective, and sends the class of the constant map $1$ to that same distinguished class. Its target is the whole cohomology set on $Q$. For a group $R$, the series condition used here means that there exist $m\in\mathbb N$ and a nondecreasing sequence $(S_i)_{i\in\mathbb N}$ of normal subgroups of $R$ such that $S_0=\{1\}$, $S_m=R$, and, for each $i<m$, some $r_i\in R$ satisfies $S_{i+1}=\langle S_i,r_i\rangle$. Repeated terms are allowed, and $m=0$ is allowed precisely when $R$ is trivial. Saying that $R$ is prosupersolvable means that every quotient $R/U$ by an open normal subgroup satisfies this series condition. The quotient groups $J/U$, their subgroup images, and their indices are finite here, so the natural-number cardinal and index conventions do not replace an infinite value by $0$. A subgroup image of order $1$ or index $1$ imposes no prime-divisor condition on that number. Trivial $J$ or $N$ are permitted; $p=0$ and $p=1$ are excluded by primality.