On the $c_0$-extension property for compact lines

We present a characterization of the continuous increasing surjections $\phi:K\to L$ between compact lines $K$ and $L$ for which the corresponding subalgebra $\phi^*C(L)$ has the $c_0$-extension property in $C(K)$. A natural question arising in connection with this characterization is shown to be independent of the axioms of ZFC.