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Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). Let $p:\tilde{X}\to X$ be the universal covering of $X$.

I have seen the following -

  1. For any $g\in G$, $\theta_g:X\to X$ is the map given by $x\mapsto g\cdot x$. Then $\theta_g$ can be covered by a homeomorphism of $\tilde{X}$ since $\tilde{X}$ is simply connected, and any two such liftings differ by a deck transformation (This question). So that we have an exact sequence - $$0\to\pi_1(X)\to G'\to G\to0$$

  2. The quotient spaces $X/G$ and $\tilde{X}/G'$ are homeomorphic. (This question)

My question - Is $G'$ unique?

More precisely, if $G''$ is a group that is an extension of $\pi_1(X)$ by $G$ and $G''$ acts on $\tilde X$ so that $X/G\cong \tilde X/G''$ then is $G'\cong G''$?

Thank you.

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