| Summary: | In this paper, we investigate properties concerning some recently introduced finite coarse shape invariants—the <i>k</i>-th finite coarse shape group of a pointed topological space and the <i>k</i>-th relative finite coarse shape group of a pointed topological pair. We define the notion of finite coarse shape group sequence of a pointed topological pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>X</mi><mo>,</mo><msub><mi>X</mi><mn>0</mn></msub><mo>,</mo><msub><mi>x</mi><mn>0</mn></msub></mfenced></semantics></math></inline-formula> as an analogue of homotopy and (coarse) shape group sequences and show that for any pointed topological pair, the corresponding finite coarse shape group sequence is a chain. On the other hand, we construct an example of a pointed pair of metric continua whose finite coarse shape group sequence fails to be exact. Finally, using the aforementioned pair of metric continua together with a pointed dyadic solenoid, we show that finite coarse-shape groups, in general, differ from both shape and coarse-shape groups.
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