Counting induced k-vertex subgraphs with automorphism group exactly Q is #W[1]-hard for every finite group Q, via clique-scaffold reductions from k-clique.
Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders , booktitle =
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Counting Small Induced Subgraphs: Hardness of Symmetry-Based Properties
Counting induced k-vertex subgraphs with automorphism group exactly Q is #W[1]-hard for every finite group Q, via clique-scaffold reductions from k-clique.