The number Col(Q)(K) of colorings of a knot K by a finite quandle Q has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement Col(Q)(F)(K) to this knot invariant involving any computable functor F from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor F = (ab) that sends each finitely presented group H to its abelianization H-ab = H/[H.H]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with <= 11 crossings.