For a universal cover $p:\widetilde{X}\to X$ we have two actions of $\pi_1(X,x_0)$ on the fiber $p^{-1}(x_0)$, namely the action given by lifting loops at $x_0$ and the action given by restricting deck transformations to the fiber. Are these two actions the same when $X=S^1\vee S^1$ or $X=S^1\times S^1$? Do the actions always agree when $\pi_1(X,x_0)$ is abelian?
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