short exact sequence from fibration

Question

Let $G$ be a finite group acting freely on a path connected topological space $X$. The covering map $X \to X/G$ induces a long exact sequence of homotopy groups. Since $\pi_1(G) = 1$ and $\pi_0(X) = 0$, we obtain a short exact sequence $$ 1 \to \pi_1(X) \to \pi_1(X/G) \to G \to 0. $$ The last map here, call it $\partial$, is in principle only a map between pointed sets.

Baby question: Is $\partial$ always a group homomorphism?

Question: If $\partial$ is a group homomorphism, then the above is a short exact sequence of groups. Is it always non-split? (Assuming of course that $G, \pi_1(X) \neq 1$. I am most interested in the case when $\pi_1(X) = \mathbf{Z}$.)

Answer 1

Let $X=S^1\times S^\infty\to X/G=S^1\times\mathbb{RP}^\infty$ where $G=\mathbb Z/2$. Then the short exact sequence is $0\to\mathbb {Z}\to\mathbb {Z}\times\mathbb{Z}/2\to\mathbb Z/2\to 0$, which is split.

Answer 2

For your main question, covering space theory tells us that every short exact sequence of groups $$1 \to N \to E \to G \to 1 $$ occurs in this fashion. So, pick your favorite non-split example with $G$ finite.

Here's a few more details. Pick a path connected CW-complex $Y$ such that $\pi_1(Y)$ is isomorphic to $E$. Covering space theory gives us the following:

• A regular covering map $f : X \mapsto Y$ such that $X$ is path connected, and such that the induced homomorphism $f_* : \pi_1(X) \to \pi_1(Y)\approx E$ is injective and has image $N$.
• An isomorphism between the quotient group $G \approx E/N$ and the deck transformation group of the covering map $f$, hence $Q$ acts freely and properly discontinuously on $S$ with quotient space $Y$.