  A kernel in a digraph is an independent and absorbent subset of its vertex
set. A digraph is critical kernel imperfect if it does not have a kernel, but
every proper induced subdigraph does. In this article, we characterize
asymmetrical $4$-quasi-transitive and $4$-transitive digraphs, as well as
$2$-anti-transitive, and asymmetrical $4$-anti-transitive digraphs with bounded
diameter, which are critical kernel imperfect.
