Suppose $H\hookrightarrow G$ is an inclusion of topological groups and $H$ is closed in $G$. Then we have an exact sequence $$ 1\to H\to G\xrightarrow \pi G/H \to 1$$ where $G/H$ is just a pointed topological space. It is easy to prove that $G\to G/H$ is a principal $H$-bundle (but not a locally trivial bundle). Denote this bundle by $\eta$.
Let $X$ be a topological space and define $H^1(X,G)$ as a pointed set of principal $G$-bundles over $X$. Is my proof of the following theorem correct?
Theorem: One has an exact sequence $$0\to C(X,H)\to C(X,G)\xrightarrow {\pi _*} C(X,G/H)\xrightarrow{\delta} H^1(X,H)$$ where $\delta (f):=f^*\eta$ and $C(X,-)$ is a pointed set of continuous functions.
Proof: we will just prove the exactness at $C(X,G/H)$, anything else seems obvious.
$1) \; Im (\pi _*)\subset \operatorname{Ker} (\delta):$ Let $f\in C(X,G)$ and we consider $\delta (\pi (f))=f^*(\pi ^* (\eta))$. Pullback of $\eta$ along $\pi$ is trivial so $f^*(\pi ^* (\eta))$ is trivial.
$2) \; Ker (\delta)\subset Im (\pi _*):$ Suppose $f\in C(X,G/H)$ and $\delta (f)=f^* (\eta )$ is a trivial bundle. Then we have a section $\alpha :X\to X\times _{G/H} G$, $\alpha (x)=(x,g_x)$ and we can define $\tilde f \in C(X,G)$ as $\tilde f (x):= g_x$. By the definition of the pullback $\pi \circ \tilde f =f$. QED
The reason I ask this is that I saw many variations of this theorem, mostly in terms of Cech cohomology, and they always require a lot of assumptions like $H$ is normal and Lie or they consider only smooth functions. So maybe I lose some crucial points here?