I am currently studying a complexity problem related to symmetries, and am considering a study of the parameterized complexity of the problem.
In theory, any part of the input can be fixed as a parameter, but in practice it seems like parameters are almost always integers. Some problems use a graph as a parameter, such as the testing whether H is a minor of G (polynomial when H is fixed, hence fixed parameter tractable with H as a parameter).
My question is: do you know any study of the complexity of a problem parameterized by a group?
Candidates could be classical group-related problems, such as coset intersection problem or hidden subgroup problem, but I have not found such parameterized complexity study for them.