Under ETH, isomorphism-invariant problems cannot be NP-complete on power graphs; Graph Motif is hard on power graphs of cyclic groups, and recognition is polynomial for abelian and some nilpotent power graphs.
On the Structure of the Power Graph and the Enhanced Power Graph of a Group
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On the Complexity of Problems on Graphs Defined on Groups
Under ETH, isomorphism-invariant problems cannot be NP-complete on power graphs; Graph Motif is hard on power graphs of cyclic groups, and recognition is polynomial for abelian and some nilpotent power graphs.