Let $C(X,\mathbb{Z})$, $C(X,\mathbb{Q})$ and $C\left(X\right)$ denote the $\ell$-groups of integer-valued, rational-valued and real-valued continuous functions on a topological space $X$, respectively. Characterizations are given for the extensions $C(X,\mathbb{Z})\le C(X,\mathbb{Q})\le C\left(X\right)$ to be rigid, major, and dense.