Example for a covering map with not equal groups from different base points

Question

I am looking for the examples of a covering map $p: E \to B$ that satisfies following:

Let $e_0$, $e_1$ be points of $p^{-1}(b_0)$ and $H_i=p_*(\pi_1(E,e_i))$

Then, $H_0$ and $H_1$ are not eqaul.

I think covering maps for $\mathbb R \to S^1$ or $S^1 \to S^1$ induce the equal groups.

What is an example of such a covering map?

Answer 1

Take $p:S^1 \sqcup S^1 \rightarrow S^1$. The map from first $S^1 $ to $S^1 $ is $z \rightarrow z^n $ and from the second $S^1 $ to $S^1 $ is identity map. This is a covering map. Now say for $1\in S^1 $ if $e_0$ is a preimage of $1$ in the first $S^1 $, the corresponding $p_*$ will just be multiplication by $n $. For the second case it will be identity.