Lemma 1.2

For every profinite group $J$ in universe $u$, every finite nilpotent group $N$ in universe $v$ with the discrete topology, and every jointly continuous action of $J$ on $N$ by group automorphisms, assume that either $N\rtimes J$ is prosupersolvable or $J$ is pronilpotent. Here the semidirect product has multiplication $(n,j)(n',j')=(n(j\cdot n'),jj')$ and the product topology. Let $D$ be the set of natural primes $p$ for which $p$ divides the number of elements of $J/U$ for some open normal subgroup $U$ of $J$, and, for each $p\in D$, choose a subgroup $P_p\le J$ that is Sylow pro-$p$ in $J$. A Sylow pro-$p$ subgroup $P$ of a subgroup $A\le J$ means a subgroup $P\le A$ that is closed in $J$, for which every quotient $P/U$ by an open normal subgroup of $P$ has the property that every element is killed by some power $p^k$ with $k\in\mathbb N$, and that is maximal under inclusion among the closed subgroups of $J$ contained in $A$ with this quotient property. 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 $1$. Let $I_J(A,N)$ be the subset of $H^1(A,N)$ consisting of classes with a representative $f$ such that, for every $j\in J$, there exists $n_j\in N$ satisfying $j\cdot f(j^{-1}xj)=n_j^{-1}f(x)(x\cdot n_j)$ for every $x\in A$ for which $j^{-1}xj\in A$. This condition is imposed only on that intersection; it does not require $j$ to normalize $A$. The distinguished element of $I_J(A,N)$ is again the class of the constant map $1$. Then the simultaneous restriction map $H^1(J,N)\to\prod_{p\in D}I_J(P_p,N)$, sending $[f]$ to $([f|_{P_p}])_{p\in D}$, is both injective and surjective, and it sends the class of the constant map $1$ to the family of such classes. The target is the full product of these subsets, with no further compatibility condition between different primes. Saying that a topological group $R$ is pronilpotent means that $R/U$ is nilpotent for every open normal subgroup $U$ of $R$, with the subgroup and quotient topologies understood. 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. Trivial $J$ or $N$ are allowed. If $D$ is empty, the product consists of the unique empty family, so the bijectivity assertion says that $H^1(J,N)$ has one element.