The notion of a free covering $(k+1)$-group of an $(sk+1)$--group was defined in [16] as a generalization of the notion of a free covering group (see [32]). In the paper various constructions of such $(k+1)$-groups are discussed and their functorial nature is emphasized. These constructions lead to the corresponding constructions of associated $(k+1)$-groups (the notion of an associated group is due to Post [32]) and in consequence to a new functor $As^s\colon Gr_{sk+1}\to Gr_{k+1}$. The main purpose of the second part of the paper is to describe some basic properties of the functor $As$.