Characterising subgroup

Question

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have that $p_{\#} \pi_1(\hat{X},\omega(0)) = \alpha \ p_{\#} \pi_1(X,x_0) \ \alpha^{-1} $. Now I found in my notes the sentence: In particular, if $\hat{X}$ is path-connected, then we have that two characteritising subgroups of $p$ are conjugated to each other. Now, I think this conjugated each other is nonsense and it was meant to mean 'the same'. Am I right about this? Cause they are always conjugated to each other ( as the theorem says).