Problem about covering space Let $p:\tilde{X}\to X$ be a covering space,  $\tilde{X}$ and $X$ are both path-connected and locally path-connected, if $p(x_1)=p(x_2)=x$, is $p_\ast(\pi_1(\tilde{X},x_1))=p_\ast(\pi_1(\tilde{X},x_2))$ always true?

Not in general. These subgroups will be conjugate but might not be equal. You can check this by taking a path between your points in $\widetilde{X}$ and projecting it down to a loop in $X$. It is true if $p$ is a regular covering.
Also, you don't need the "locally path connected" assumption for any of these arguments.
