  We construct an exactly solvable commuting projector model for a $4+1$
dimensional ${\mathbb Z}_2$ symmetry-protected topological phase (SPT) which is
outside the cohomology classification of SPTs. The model is described by a
decorated domain wall construction, with "three-fermion" Walker-Wang phases on
the domain walls. We describe the anomalous nature of the phase in several
ways. One interesting feature is that, in contrast to in-cohomology phases, the
effective ${\mathbb Z}_2$ symmetry on a $3+1$ dimensional boundary cannot be
described by a quantum circuit and instead is a nontrivial quantum cellular
automaton (QCA). A related property is that a codimension-two defect (for
example, the termination of a ${\mathbb Z}_2$ domain wall at a trivial
boundary) will carry nontrivial chiral central charge $4$ mod $8$. We also
construct a gapped symmetric topologically-ordered boundary state for our
model, which constitutes an anomalous symmetry enriched topological phase
outside of the classification of arXiv:1602.00187, and define a corresponding
anomaly indicator.
