The article substantiates a method for solving linear Diophantine equations using the theory of group actions. The purpose of this paper is to introduce actions of certain groups on the set of linear Diophantine equations and to study their properties related to the set of solutions of these equations. Using group-theoretic methods, we achieve the goal and establish that the actions of symmetry groups of regular $n$-dimensional polyhedra on the set of equations under study are reduced to a combination of the actions of the symmetric group $S_n$ and the automorphism group of the group of integers $Aut(\mathbb{Z})$ on the same set. The relationship between the actions of a group of parallel transfers on the set of linear Diophantine equations and on the set of their solutions is also studied: for example, the vector of the general solution of an equation obtained as a result of an action can be found as the sum of the vector of the general solution of the equation that was subjected to the action and the vector of parallel transfer. In this article, we continued the formation of a class of linear Diophantine equations. Thus, it became possible to solve more equations using the solution of just one representative.