Some examples of invariably generated groups

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