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Given a countable discrete group $G$ and suppose $G$ acts on a compact metrizable abelian group $Y$ with normalized Haar measure $\mu$, measure preserving, let $\mathbb{T}$ denotes the unite circle.

Assume $X$ is a closed subgroup of $Y$ and denote $\nu$ to be the normalized Haar measure on $X$ And assume $GX\subset X$.

We may consider a measurable cocycle $c: G\times X\to \mathbb{T}$. Recall that a cocycle satisfies $c(st,x)=c(s,tx)c(t,x), \forall s,t\in G, x\in X$.

My question is:

Under what conditions can $ c$ be extended to a cocycle from $G\times Y$ to $\mathbb {T}$?

Some trivial observations:

1, when $c$ is a trivial cocycles, I.e., it does not depend on $X$, it extends.

2, when $Y=X\times X$, it does.

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