Is there a group whose manifold is a fiber bundle with base is $S_1$ and fiber $\mathbb{Z_2}?$

Question

Let's consider a fiber bundle with base $S_1$ and fiber $\mathbb{Z}_2$. I want this manifold to be topologically non-trivial, the edge of the Möbius strip.

How do I know if is it possible to introduce a group structure on such a manifold? So that the manifold would turn into a principle bundle.

Answer 1

This manifold, considered independent of its fiber bundle structure, is just $S^1$ again, so it admits the same group structure as $S^1$. You can think of the resulting bundle as the short exact sequence

$$1 \to \mathbb{Z}_2 \to S^1 \xrightarrow{x \mapsto x^2} S^1 \to 1.$$