The GIT-fan assigned to an algebraic variety with the action of an algebraic group is a polyhedral fan enumerating all GIT-quotients in the sense of Mumford. We discuss an algorithm to compute the GIT-fan for torus actions on affine varieties with finite symmetries, and its implementation in OSCAR. Relying on Gröbner basis techniques to decide monomial containment, computations with rational polyhedral cones, and expanding orbits with respect to actions of permutation groups, the algorithm combines computational methods from commutative algebra, convex geometry and group theory. Consequently, OSCAR, for the first time, provides an open-source framework that seamlessly manages every stage of the algorithm. We illustrate the algorithm through examples.