Between the enhanced power graph and the commuting graph

The purpose of this note is to define a graph whose vertex set is a finite group G $G$, whose edge set is contained in that of the commuting graph of G $G$ and contains the enhanced power graph of G $G$. We call this graph the deep commuting graph of G $G$. Two elements of G $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of G $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that automorphisms of G $G$ act as automorphisms of the deep commuting graph.