Classification of non-solvable groups whose power graph is a cograph

In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They clas-sify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL2(q) or Sz(2(2e+1)). In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL2(q) or Sz(2(2e+1)). For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We ob-tain a complete classification of solvable power-cograph groups whose Gruenberg-Kegel graph is connected. Moreover, we reduce the case where the Gruenberg-Kegel graph is disconnected to the classification of p-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.