So far, in the studies of finite subgroups of Cremona groups, they were all considered on
an equal footing. However, in reality it is necessary to consider some of them as “not basic”,
since they are obtained from others by a standard “base change” construction. This leads to
the problem of finding the subgroups in the classification lists, which are obtained by such
a nontrivial change, or, in another terminology, are nontrivially compressible. The talk is
aimed to discuss this topic.