Compressing finite subgroups of Cremona groups

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.