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) -> C(L). We show that if such an operator exists then there is one which norm 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 subset X of [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.
Based on the recent preprint https://arxiv.org/abs/2305.04565