A workflow specification defines sets of steps and users. An authorization policy imposes constraints on which users may perform particular steps. Other security requirements, such as separation-of-duty, impose constraints on which groups of users may perform sets of steps. The \emph{workflow satisfiability problem} (WSP) is the problem of determining whether there exists an assignment of users to workflow steps that satisfies all such constraints. An algorithm for solving WSP is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. Given the difficulty of WSP, it is important, particularly for the second application, that such algorithms are as efficient as possible. A constraint is said to be \emph{user-independent} if the identities of users are irrelevant to the satisfaction of a constraint. Recent work has developed fixed-parameter tractable algorithms for solving WSP, under the assumption that all constraints are user-independent. In this paper, we generalize the notion of a user-independent constraint to a \emph{class-independent} constraint, enabling us to model scenarios where the set of users may be partitioned into user groups. We prove that solving WSP remains fixed-parameter tractable for this more general class of constraints and develop an algorithm derived from our results. We compare the performance of our bespoke algorithm with that of SAT4J (a pseudo-Boolean SAT solver). Our results demonstrate that our algorithm is more suitable than SAT4J for many instances of WSP.