  We investigate the problem of existence of a bounded extension to $C(K)$ of a
bounded $c_0(I)$-valued operator $T$ defined on the subalgebra of $C(K)$
induced by a continuous increasing surjection $\phi:K\to L$, where $K$ and $L$
are compact lines. Generalizations of some of the results of [6] about
extension of $c_0$-valued operators are obtained. For instance, we prove that
when a bounded extension of $T$ exists then an extension can be obtained with
norm at most twice the norm of $T$. Moreover, the class of compact lines $L$
for which the $c_0$-extension property is equivalent to the $c_0(I)$-extension
property for any continuous increasing surjection $\phi:K\to L$ is studied.
