  The semidirect product of a finitely generated group dual with the symmetric
group can be described through so-called group-theoretical categories of
partitions (covers only a special case; due to Raum--Weber, 2015) and skew
categories of partitions (more general; due to Maassen, 2018). We generalize
these results to the case of graph categories, which allows to replace the
symmetric group by the group of automorphisms of some graph.
