I have been studying Algebraic Topology from Allen Hatcher (pg.67) and I am confused with a statement in the text; first I shall give some background.
We know that:
If $X$ is path-connected, locally path-connected space, then two path connected covering spaces $p_1:\widetilde{X}_1\rightarrow X,p_2:\widetilde{X}_2\rightarrow X,$ are isomorphic via the isomorphism $f:\widetilde{X}_1\rightarrow\widetilde{X}_2$ such that $\widetilde{x}_1\in p^{-1}(x_0)\mapsto\widetilde{x}_2\in p^{-1}(x_0)$ if and only if $p_{1_*}(\pi_1(\widetilde{X}_1,\widetilde{x}_1))=p_{2_*}(\pi_1(\widetilde{X}_2,\widetilde{x}_2)).$
Now the next theorem says:
Theorem: Let $X$ be path-connected, locally path-connected, and semilocally simply-connected. Then there is a bijection between the set of basepoint-preserving isomorphism classes of path-connected covering spaces $p:\left(\widetilde{X}, \tilde{x}_{0}\right) \rightarrow\left(X, x_{0}\right)$ and the set of subgroups of $\pi_{1}\left(X, x_{0}\right),$ obtained by associating the subgroup $p_{*}\left(\pi_{1}\left(\tilde{X}, \tilde{x}_{0}\right)\right)$ to the covering space $\left(\widetilde{X}, \tilde{x}_{0}\right) .$ If basepoints are ignored, this correspondence gives a bijection between isomorphism classes of path-connected covering spaces $p: \widetilde{X} \rightarrow X$ and conjugacy classes of subgroups of $\pi_{1}\left(X, x_{0}\right).$
The first statement is no issue, it follows quite easily; the proof of the second starts as -
We show that for a covering space $p:\left(\widetilde{X}, \tilde{x}_{0}\right) \rightarrow\left(X, x_{0}\right),$ changing the basepoint $\tilde{x}_{0}$ within $p^{-1}\left(x_{0}\right)$ corresponds exactly to changing $p_{*}\left(\pi_{1}\left(\widetilde{X}, \tilde{x}_{0}\right)\right)$ to a conjugate subgroup of $\pi_{1}\left(X, x_{0}\right) .$
The proof of this makes sense to me, but I just don't see how this is answering the proof required and asked in the Theorem. I know I have to use the first result, but I cannot wrap my head around what's happening with and without the fixing of basepoints. Any help will be very appreciated.