In this paper we propose a new class of contrapositivisation operators for fuzzy implications called (QO,QG,N)-contrapositivisators and whose generators are quasi-overlaps, quasi-groupings and fuzzy negations. We show that (QO,QG,N)-contrapositivisators generalize medium contrapositivisators and (G,N)-contrapositivisators. We present a brief study on the notion of N-compatibility for (QO,QG,N)-contrapositivisations and show that the class of (QO,QG,N)-contrapositivisators is invariant by the action of automorphisms. We extensively study the characterizations of (QO,QG,N)-contrapositivisators with respect to the main properties that are generally satisfied by fuzzy implications. This study was done from two points of view: first, considering that the input implication is arbitrary and second, assuming that the universe of the input implications consists of four specific classes of fuzzy implications generated from overlaps and/or groupings. In addition, we present new methods to construct quasi-overlap (overlap) and quasi-grouping (grouping) functions from (QO,QG,N)-contrapositivisators.