  Cameron, Manna and Mehatari investigated the question of which finite groups
admit a power graph that is a cograph, also called power-cograph groups
(Journal of Algebra 591 (2022)). They give a classification for nilpotent
groups and partial results for general groups. However, the authors point out
number theoretic obstacles towards a classification. These arise when the
groups are assumed to be isomorphic to PSL 2 (q) or Sz(q) and are likely to be
hard. 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 PSL 2 (q) or Sz(q).
For the solvable case, we are able to precisely describe the structure of
solvable power-cograph groups. We obtain 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.
