  Invariable generation is a topic that has predominantly been studied for
finite groups. In 2014, Kantor, Lubotzky, and Shalev produced extensive tools
for investigating invariable generation for infinite groups. Since their paper,
various authors have investigated the property for particular infinite groups
or families of infinite groups.
  A group is invariably generated by a subset $S$ if replacing each element of
$S$ with any of its conjugates still results in a generating set for $G$. In
this paper we investigate how this property behaves with respect to wreath
products. Our main work is to deal with the case where the base of $G\wr_X H$
is not invariably generated. We see both positive and negative results here
depending on $H$ and its action on $X$.
