Problem with a group as complexity parameter? 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.
Many of the best results around graph isomorphism and permutation group algorithms are parametrized by some structure of a relevant group. For example, let $\mathcal{S}_d$ denote the class of groups all of whose composition factors are isomorphic to subgroups of $S_d$. Then coset intersection for groups in $\mathcal{S}_d$ can be solved in polynomial time for constant d. This is closely related to the polynomial-time isomorphism test for bounded-degree graphs. 

I think there are also several other problems that are solvable in polynomial time when certain associated groups are in $\mathcal{S}_d$ for $d=O(1)$, but I don't have references off the top of my head.