  A graph $G$ is $H$-free if it does not contain an induced subgraph isomorphic
to $H$. The study of the typical structure of $H$-free graphs was initiated by
Erd\H{o}s, Kleitman and Rothschild, who have shown that almost all $C_3$-free
graphs are bipartite. Since then the typical structure of $H$-free graphs has
been determined for several families of graphs $H$, including complete graphs,
trees and cycles. Recently, Reed and Scott proposed a conjectural description
of the typical structure of $H$-free graphs for all graphs $H$, which extends
all previously known results in the area.
  We construct an infinite family of graphs for which the Reed-Scott conjecture
fails, and use the methods we developed in the prequel paper to describe the
typical structure of $H$-free graphs for graphs $H$ in this family.
  Using similar techniques, we construct an infinite family of graphs $H$ for
which the maximum size of a homogenous set in a typical $H$-free graph is
sublinear in the number of vertices, answering a question of Loebl et al. and
Kang et al.
