  We consider the family of gapped Hamiltonians introduced in [FNW] on the
quantum spin chains $\bigotimes_{\mathbb Z}\mathop{\mathrm{M}}\nolimits_n$,
with local symmetry given by a group $G$. The $G$-symmetric gapped Hamiltonians
are given by triples $(k,u,V)$, where $u$ is a projective unitary
representation of $G$ on a finite dimensional space ${\mathbb C}^k$, and $V$ is
an isometry from ${\mathbb C}^k$ to ${\mathbb C}^n\otimes {\mathbb C}^k$. We
show that Hamiltonians $H_0,H_1$, given by the triples
  $(k^{(0)}, u^{(0)}, V^{(0)})$ and $(k^{(1)}, u^{(1)}, V^{(1)})$ are
$C^1$-equivalent if the projective representations $u^{(0)}$ and $u^{(1)}$ are
unitary equivalent.
