  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.
