Weighted interval set scheduling problem Given a set of weighted intervals, the weighted interval scheduling problem is to select a subset of the intervals such that none of the intervals in the subset overlap and the sum of their weights is maximized. This can be solved in linear time with dynamic programming or memoized recursion.

Is there a version of this problem where intervals are grouped, i.e. interval $a$ can only be in the solution if intervals $b$ and $c$ are also in the solution? By this formulation, there would be a weight associated with each group, or set, of intervals.

The most similar problem I've come across is the exact cover problem, which is NP-Complete, but in this instance we're maximizing weight and all vertices need not be covered.

Any insight will be much appreciated. Thanks!
Sure, you can define any problem you want.  Effectively you would no longer seek a cover of intervals, but a cover of sets, where each set is a union of intervals.  Over a discrete set, any set can be expressed as a union of intervals.  Thus, this is equivalent to the maximum set packing problem.  Consequently, it is NP-hard.