Group action on the fundamental group of a covering space?

Question

I can see that I'd have to show:

For each $\tilde x \in Y, [g] \in p_* \pi_1(\tilde X, \tilde x)$ iff $\phi([g]) = 1_{S_Y}$ but I'm having trouble with both directions.

I would assume that defining a lifting as $\tilde x [f]$ may be necessary but I'm not sure how I would make it work.

Anyone have any ideas?

Answer 1

If $[f]\in\ker\theta$, then $[f]$ lifts to a loop at any basepoint.

For the converse, note that $p_*\pi_1(\widetilde{X},\tilde{x})$ stabilizes $\tilde{x}$. The intersection of all the stabilizers surely stabilizes all of $Y$, and hence induces the trivial permutation.