  A permutation snark is a snark which has a 2-factor $F_2$ consisting of two
chordless circuits; $F_2$ is called the permutation 2-factor of $G$. We
construct an infinite family $\mathcal H$ of cyclically 5-edge connected
permutation snarks. Moreover, we prove for every member $G \in \mathcal H$ that
the permutation 2-factor given by the construction of $G$ is not contained in
any circuit double cover of $G$.
