I am reading Serre's paper "Sur les groupes de congruence des variétés abéliennes" (here is the link to this paper: https://www.mathnet.ru/links/016949238724700ec2209f00e507a40f/im3061.pdf). I am stuck at Proposition 7 which showed that the first Galois cohomology commutes with projective limits when the coefficients are compact. Here is the setting of the result.
Proposition 7: Let $G$ be a profinite group and $A=\displaystyle \lim_{\leftarrow}A_n$ be the projective of compact $G$-modules. Then the canonical map $H^1(G,A)\to\displaystyle\lim_{\leftarrow}H^1(G,A_n) $ is bijective.
To prove the surjectiveness, he choosed a family of 1-cycles $(x_n)$ on the right side, and for each $x_n$, let $Z_n$ be the set of 1-cocycle in the class of $x_n$. Then he said that $Z_n$ is a homogenous space over $A_n$, and the stabilizer of any element in $Z_n$ is $A_n^G$.
I do not understand how he equipped the homogenous space structure on $Z_n$? Could you help me to clarify this step? I believe the action of $A_n$ on $x_n$ is given by $$x_n\cdot a:=x_n.\sigma_a $$ where $\sigma_a(g)=\dfrac{ga}{a}$, but I do not know how $G$ acts on $Z_n$.
I know that in the finite coefficients cases, there is systematic proof of Tate in the paper "Relations between K2 and Galois cohomology" ( or in Neukirch's book - Cohomology of Number Fields, Chapter II $7), but the proof of Serre is very wonderful and intrinsic.
Thank you very much!