  Although the notion of crossed groups was originally introduced only in the
simplicial case, the definition makes sense in the other categories. For
instance, Batanin and Markl studied crossed interval groups to investigate
symmetries on the Hochschild cohomology in view of operads. The aim of this
paper is to make a comprehensive understanding of crossed groups for arbitrary
base categories. In particular, we focus on the local presentability of the
category of crossed groups, monadicity, and the basechange theorem along
certain sorts of functors. The paper also contains the classification of
crossed interval groups, which Batanin and Markl concerned about.
