Countable discrete extensions of compact lines

We consider a separable compact line $K$ and its extension $L$ consisting of $K$ and a countable number of isolated points. The main object of study is the existence of a bounded extension operator $E:C(K)\to C(L)$. We show that if such an operator exists then there is one for which $\mathrm{\parallel }E\mathrm{\parallel }$ is an odd natural number. We prove that if the topological weight of $K$ is bigger than or equal to the least cardinality of a set $X\subseteq [0,1]$ that cannot be covered by a sequence of closed sets of measure zero then there is an extension $L$ of $K$ admitting no bounded extension operator.