For a finite group, we present three algorithms to compute a generating set of invariant simultaneously to generating sets of basic equivariants, i.e., equivariants for the irreducible representations of the group. The first construction applies solely to reflection groups and consists in applying symmetry preserving interpolation, as developed by the same authors, along an orbit in general position. The second algorithm takes as input primary invariants and the output provides not only the scondary invariants but also free bases for the modules of basic equivariants. The third algorithm proceeds degree by degree, determining the fundamental invariants as forming the H-basis of the ideal of the Nullcone, and the fundamental equivariants as constituting the symmetry adapted basis of an invariant complement. Remarkably, the here presented algorithms are the very first algorithms to compute both the fundamental invariants and the fundamental equivariants simultaneously. Fundamental equivariants allow to assemble symmetry adapted bases of higher degrees, and these are essential ingredients in exploiting and preserving symmetry in computations. They appear within algebraic computation and beyond, in physics, chemistry and engineering.