  Motivated by the search for idempotent cellular automata (CA), we study CA
that act almost as the identity unless they read a fixed pattern $p$. We show
that constant and symmetrical patterns always produce idempotent CA, and we
characterize the quasi-constant patterns that produce idempotent CA. Our
results are valid for CA over an arbitrary group $G$. Moreover, we study the
semigroup theoretic natural partial order defined on idempotent CA. If $G$ is
infinite, we prove that there is an infinite independent set of idempotent CA,
and if $G$ has an element of infinite order, we prove that there is an infinite
increasing chain of idempotent CA.
