On subgroups which cover or avoid chief factors of a finite group

A classical topic of research in Finite Group Theory is the following:
What is the influence on the structure of the group of the fact that all members of some relevant family of subgoups enjoy a given embedding property?
The cover-avoidance property is a subgroup embedding property that has recovered much attention in the last few years. In this survey article we present a number of results showing that its influence goes much further from the classical supersolubility.