  The Houghton groups $H_1, H_2, \ldots$ are a family of infinite groups. In
1975 Wiegold showed that $H_3$ was invariably generated (IG) but $H_1\le H_3$
was not. A natural question is then whether the groups $H_2, H_3, \ldots$ are
all IG. Wiegold also ends by saying that, in the examples he had found of an IG
group with a subgroup that is not IG, the subgroup was never of finite index.
Another natural question is then whether there is a subgroup of finite index in
$H_3$ that is not IG. In this note we prove, for each $n\in \{2, 3, \ldots\}$,
that $H_n$ and all of its finite index subgroups are IG.
  The independent work of Minasyan and Goffer-Lazarovich in June 2020 frames
this note quite nicely: they showed that an IG group can have a finite index
subgroup that is not IG.
