  We investigate the following problem posed by Cabello Sanch\'ez, Castillo,
Kalton, and Yost:
  Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial
twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$
containing a non-complemented copy $Z$ of $c_0$ such that the quotient space
$X/Z$ is isomorphic to $C(K)$?
  Using additional set-theoretic assumptions we give the first examples of
compact spaces $K$ providing a negative answer to this question. We show that
under Martin's axiom and the negation of the continuum hypothesis, if either
$K$ is the Cantor cube $2^{\omega_1}$ or $K$ is a separable scattered compact
space of height $3$ and weight $\omega_1$, then every twisted sum of $c_0$ and
$C(K)$ is trivial. We also construct nontrivial twisted sums of $c_0$ and
$C(K)$ for $K$ belonging to several classes of compacta. Our main tool is an
investigation of pairs of compact spaces $K\subseteq L$ which do not admit an
extension operator $C(K)\to C(L)$.
