Exact discrete symmetries, if non-linearly realized, can reduce the ultraviolet sensitivity of a given theory. The scalars stemming from spontaneous symmetry breaking are massive without breaking the discrete symmetry, and those masses are protected from divergent quadratic corrections. This is in contrast to non-linearly realized continuous symmetries. The symmetry-protected masses and potentials of those discrete Goldstone bosons offer promising physics avenues, both theoretically and in view of the blooming experimental search for ALPs. In this text, we develop this theoretical setup for the specific case of a triplet of $A_5$ using invariant theory, showcasing the substantial improvements and compelling phenomenological consequences introduced by the invariance under a discrete symmetry.