Classification of double coverings and $H^1(X ; \mathbb{Z}/2\mathbb{Z})$

It is a well known fact (see for example Hatcher’s “Algebraic Topology”, chapter $ 1$ ) that there is a bijection between the $ n$ -sheeted coverings of $ X$ up to isomorphism of covering spaces and the conjugacy classes of homomorphism between $ \pi_1(X)$ and $ S_n$ (where $ S_n$ is the symmetric group), under the very poor assumption that $ X$ admits universal covering. The proof of this fact (at least, the one presented in Hatcher’s book) is quite techincal, although not transcendental.

If we focus our attention on double coverings, things simplify quite a lot, and we have a bijection between the double coverings of $ X$ and $ Hom(\pi_1(X), S_2)$ ; by universal coefficient theorem, we have so a bijection $ $ \{Double\ coverings\ of\ X\} \longleftrightarrow H^1(X; \mathbb{Z}/2\mathbb{Z}).$ $ Does exist a simpler way to prove this (natural) bijection in this particular case?