I would like to seek some clarification to the solution to the following problem from Representations and Characters of Groups (Gordon, Liebeck):
Let $\rho$ be a representation of the finite group $G$ over $\mathbb{C}.$ Then $\delta:g \to \text{det}\bigl(\rho(g)\bigr)$ ($g \in G$) is a linear character. If $\delta(g)=-1,$ for some $g\in G,$ show $G$ has a normal subgroup of index 2.
The solution provided by the author claims that $\text{Im} \delta $ has even order and hence contains a subgroup of index $2.$ May I know why it contains subgroup of index $2 \ ?$ Isn't $A_4$ a counterexample or am I missing some relevant context?
Indeed $A_4$ is a finite group of even order with no subgroup of index $2$.
However in your question $\operatorname{Im} \delta$ is a finite subgroup of the multiplicative group of $\mathbb{C}$, and thus $\operatorname{Im} \delta$ is cyclic.
So you are done once you observe that a finite cyclic group of even order has a subgroup of index $2$.