Determine the group of automorphisms for a covering

Question

I have a list of coverings $(\overline{X},\pi)$ of certain spaces. I'm asked to determine the group of automorphisms of the covering. I don't know clearly how to proceed.

Example

Take the following two very easy examples:

$\rho:\mathbb{R} \to \mathbb{S}^1$ given by $\rho(t) = e^{2\pi i t}$

$\rho \times \tau_n:\mathbb{R} \times \mathbb{S}^1 \to \mathbb{S}^1 \times \mathbb{S}^1$ such that $\tau(z) = z^n$.

Could you sketch the strategy to solve this examples in general?

Answer 1

For the first, an homeomorphism of $\mathbb{R}$ is a covering if and only if $\rho(F(t))=\rho(t)$. This implies that $F(t)=t+2\pi n(t)$ where $n(t)$ is an integer. Since $F$ is continuous, $n(t)$ is constant. So $F(t)=t+ 2\pi n$.