I am struggling to figure out what is wrong with my following conclusion:
Let $A \overset{f}\longrightarrow B$ be a principal $F$-bundle and $B \overset{g}\longrightarrow C$ be a principal $G$-bundle. Now by definition of a principal bundle, we know that for every point $c$ in $C$ there exists a neighborhood $U$ of $c$ such that $g^{-1}(U) \cong U \times G$. If we do the same for $f$ using neighborhoods obtained from $g$, we get $$f^{-1} \circ g^{-1}(U) \cong U \times F \times G$$ This means that $A \overset{g \circ f}\longrightarrow C$ is a principal $F \times G$-bundle.In other words, above conclusion tells us that composition of two mentioned principal bundles just comes from the following trivial central extension $$1 \to F \to F \times G \to G \to 1$$ But we know that there are non-trivial central extensions of $G$ by $F$, so we can have compositions of principal $G$-bundles with principal $F$-bundles which are not just $F \times G$-bundles.