Some examples of invariably generated groups

A group $G$ is invariably generated (IG) if there is a subset $S\subseteq G$ such that for every subset $S^{\mathrm{\prime }}\subseteq G$, obtained from $S$ by replacing each element with a conjugate, $S^{\mathrm{\prime }}$ 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 $N \lhd G$suchthat$N$ 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.