  We consider SPT-phases with on-site $G$ (where $G$ is any finite group)
symmetry for two-dimensional quantum spin systems. We then impose translation
invariance in one direction and observe that on top of the
$H^3(G,\mathbb{T})$-valued index constructed in \cite{ogata2021h3gmathbb}, an
additional $H^2(G,\mathbb{T})$-valued index emerges. We also show that if we
impose translation invariance in two directions, on top of the expected
$H^3(G,\mathbb{T})\oplus H^2(G,\mathbb{T})\oplus H^2(G,\mathbb{T})$ valued
index, an additional $H^1(G,\mathbb{T})$-valued index emerges.
