We consider the following problem closely related to graph isomorphism. In a simplified version, the task is to compute the automorphism group of a given set family (or a hypergraph), that is, the group of all automorphisms of the given sets which are compatible with some permutation of their elements. In a general setting, the set family in question is a collection of cliques (called marked cliques) of a given interval graph, and the task is to compute the group of all permutations of the cliques which result from some automorpism of the underlying interval graph. This problem is obviously at least as hard as the graph isomorphism (GI-hard) already in the simplified version -- consider the set family of edges of a graph, and we give an FPT-time algorithm parameterized by the maximum number of sets in the family which are incomparable by inclusion (its antichain size). To our best knowledge, the general version of the problem has not been formulated in the literature so far. The problem has been inspired by the research of special cases of the isomorphism problem of chordal graphs; namely, the simplified set-family version is the core of our FPT algorithm for the isomorphism of so-called Sd-graphs [MFCS 2021], and the general version extends and improves a cumbersome technical step in our FPT algorithm for the isomorphism of chordal graphs of bounded leafage [WALCOM 2022]. The new algorithm combines two classical tools -- PQ-trees of interval graphs and Babai's tower-of-groups, in a nontrivial way.