Proposition 2.1

For every profinite group $J$ in universe $u$ and discrete topological group $N$ in universe $v$ equipped with a jointly continuous action of $J$ by group automorphisms, suppose every finite subset of $N$ generates a finite subgroup. Let $p,p_0$ be distinct natural primes, suppose that for every $n\in N$ there exists $k\in\mathbb N$ with $n^{p^k}=1$, and let $J_0,Q\le J$ be closed subgroups with $J_0$ normal in $J$. Assume that some $q_0\in Q$ has its integer powers dense in $Q$, and that every quotient $Q/U$ by an open normal subgroup $U$ of $Q$ has every element killed by some power of $p_0$. Let $f:J_0\to N$ be a continuous map satisfying $f(xy)=f(x)(x\cdot f(y))$ for all $x,y\in J_0$. Suppose that for every $q\in Q$ there exists $n_q\in N$ such that $q\cdot f(q^{-1}xq)=n_q^{-1}f(x)(x\cdot n_q)$ for every $x\in J_0$ with $q^{-1}xq\in J_0$. Then there exists a continuous map $g:J_0\to N$ satisfying $g(xy)=g(x)(x\cdot g(y))$ and all three following conditions: there is one $n\in N$ with $g(x)=n^{-1}f(x)(x\cdot n)$ for every $x\in J_0$; for every $q\in Q$ and $x\in J_0$ with $q^{-1}xq\in J_0$, one has $q\cdot g(q^{-1}xq)=g(x)$; and $g(q)=1$ for every $q\in Q\cap J_0$. Normality of $J_0$ ensures that the conjugate-membership condition in these identities always holds for $x\in J_0$. The finite-generation hypothesis permits $N$ to be infinite and includes the empty generating set; no uniform bound on the exponents $k$ is required. Trivial $N$, $J_0$, or $Q$ are permitted, a dense cyclic subgroup may be trivial, and the assumptions that $p,p_0$ are prime exclude $0$ and $1$.