Given a Hilbert space and a finite family of operators defined on the space, the common fixed point problem (CFPP) is the problem of finding a point in the intersection of the fixed point sets of these operators. A particular case of the problem, when the operators are orthogonal projections, is the convex feasibility problem which has numerous applications in science and engineering. In a previous work [Censor, Reem, and Zaknoon, A generalized block-iterative projection method for the common fixed point problem induced by cutters, J. Global Optim. 84 (2022), 967--987] we studied a block-iterative method with dynamic weights for solving the CFPP assuming the operators belong to a wide class of operators called cutters. In this work we continue the study of this algorithm by allowing extrapolation, namely we weaken the assumption on the relaxation parameters. We prove the convergence of this algorithm when the space is finite dimensional in two different scenarios, one of them is under a seems to be new condition on the weights which is less restrictive than a condition suggested in previous works. Along the way we obtain various new results of independent interest related to cutters, some of them extend, generalize and clarify previously published results.