Let $G$ be a group and let $\mathcal{L}(G)$ denote the complete lattice of subgroups of $G$. We have that every automorphism of $G$ induces a lattice-automorphism on $\mathcal{L}(G)$. From here we see that every maximal subgroup of $G$ is sent to a maximal subgroup of $G$. Considering the infimum (or intersection) of all the maximal subgroups yields a characteristic subgroup of $G$ which is commonly called the Frattini subgroup. Dually, we can see that every minimal subgroup of $G$ is also sent to a minimal subgroup. Considering the supremum of all the minimal subgroups of $G$ yields another characteristic subgroup of $G$.
I have two questions. First, whether there are any references which consider this specific subgroup, as I haven't been able to find any literature on it.
Second and more pertinent, I was wondering if anyone could find a characterization of this subgroup which reflects the characterization of the Frattini subgroup. That is, the Frattini subgroup can be defined as the subgroup of all non-generators; does there exist a similar property for this other subgroup?
Letting $\Psi(G)$ represent this specific subgroup, I've found that if $A\subseteq G$ is another subset of $G$ such that $\Psi(G)\not\subseteq A$ and $\langle A, \Psi(G)\rangle = G$ then $\langle A\rangle\neq G$. But I'm looking for a characterization in terms of the elements of $\Psi(G)$.