Finitarily linear wreath products

We consider faithful finitary linear representations of (generalized) wreat h products A wr(Omega) H of groups A by H over (potentially) infinite-dimen sional vector spaces, having previously considered completely reducible suc h representations in an earlier paper. The simpler the structure of A the m ore complex, it seems, these representations can become. If A has no non-tr ivial abelian normal subgroups, the conditions we present are both necessar y and sufficient. They imply, for example, that for such an A, if there exi sts such a representation of the standard wreath product A wr H of infinite dimension, then there already exists one of finite dimension.