  Given the large class of groups already known to be sofic, there is seemingly
a shortfall in results concerning their permanence properties. We address this
problem for wreath products, and in particular investigate the behaviour of
more general metric approximations of groups under wreath products.
  Our main result is the following. Suppose that $H$ is a sofic group and $G$
is a countable, discrete group. If $G$ is sofic, hyperlinear, weakly sofic, or
linear sofic, then $G\wr H$ is also sofic, hyperlinear, weakly sofic, or linear
sofic respectively. In each case we construct relevant metric approximations,
extending a general construction of metric approximations for $G\wr H$ that
uses soficity of $H$.
