  Let $Z_{r, R}$ be the space of continuous functions on the annulus $B_{r, R}$
in $\mathbb C^n$ whose $\lambda$-twisted spherical mean, in the set up of the
M\'{e}tivier group, vanishes over the spheres $S_s(z)\subset B_{r, R} $ with
ball $B_r(0)\subseteq B_s(z).$ We characterize the spherical harmonic
coefficients of functions in $Z_{r, R},$ eventually, in terms of polynomial
growth, by which we infer support theorem. Further, we prove that non-harmonic
complex cone and the boundary of a bounded domain are sets of injectivity for
the $\lambda$-twisted spherical means.
