  A group $G$ is invariably generated (IG) if there is a subset $S \subseteq G$
such that for every subset $S' \subseteq G$, obtained from $S$ by replacing
each element with a conjugate, $S'$ generates $G$. $G$ is finitely invariably
generated (FIG) if, in addition, one can choose such a subset $S$ to be finite.
  In this note we construct a FIG group $G$ with an index $2$ subgroup $N \lhd
G$ such that $N$ is not IG. This shows that neither property IG nor FIG is
stable under passing to subgroups of finite index, answering questions of
Wiegold and Kantor, Lubotzky, Shalev. We also produce the first examples of
finitely generated IG groups that are not FIG, answering a question of Cox.
