Minimal invariable generating sets

A subset $S$ of a group $G$ invariably generates $G$ if, when each element of $S$ is replaced by an arbitrary conjugate, the resulting set generates $G\mathrm{.}$ An invariable generating set $X$ of $G$ is called minimal if no proper subset of $X$ invariably generates $G\mathrm{.}$ We will address several questions related to the behaviour of minimal invariable generating sets of a finite group.