In previous papers, the authors introduced and characterized a class of matrices called {K, s +1}-potent. Also, they established a method to construct these matrices. The purpose of this paper is to solve the associated inverse problem. Several algorithms are developed in order to find all involutory matrices K satisfying KA(s+1)K = A for a given matrix A is an element of C-nxn and a given natural number s. The cases s = 0 and s >= 1 are separately studied since they produce different situations. In addition, some examples are presented showing the numerical performance of the methods. (C) 2016 Elsevier B.V. All rights reserved.