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The spectrum of a Gelfand pair $(K\ltimes N, K)$, where $N$ is a nilpotentgroup, can be embedded in a Euclidean space. We prove that in general, theSchwartz functions on the spectrum are the Gelfand transforms of Schwartz$K$-invariant functions on $N$. We also show the converse in the case of theGelfand pair $(SO(3)\ltimes N_{3,2}, SO(3))$, where $N_{3,2}$ is the freetwo-step nilpotent Lie group with three generators. This extends recent resultsfor the Heisenberg group.

Gelfand transforms of SO(3)-invariant Schwartz functions on the free group N_{3,2}