Exercise 1, § 3

Let $G$ be the topological group $O(2,\mathbf{R})$. We have $\pi_0(G)\simeq \mathbf{Z}/2\mathbf{Z}$ and $\pi_1(G)\simeq \mathbf{Z}$; the nontrivial element of $\pi_0(G)$ acts on $\pi_1(G)$ by the mapping $t\mapsto -t$.