Consider a d (spatial) dimensional SPT phase with an on-site symmetry G, classified by some non-trivial cocycle $\omega^{d+1}(\{g_i\}) \neq \delta \mu^d(\{g_i\})$, $[\omega^{d+1}(\{g_i\})] \in H^{d+1}(G,U(1))$, . In a recent paper, Wang, Wen and Witten construct gapped boundaries via a suitable group extension $1 \longrightarrow H \overset{i}{\longrightarrow} K \overset{r}{\longrightarrow} G \longrightarrow 1 $ such that the cocycle fro $H$ defined via pullback is trivial $r^*\omega^{d+1}(\{h_i\}) = \omega^{d+1}(\{r(h_i)\})= \delta \mu^d(\{h_i\})$. The gapped boundary corresponds to an $H$ invariant theory but with $K$ gauged so that the global symmetry is $H/K \cong G$ as required. The following sentence they say however confuses me:

" By definition, two states in two different $G$-SPT phases cannot smoothly deform into each other via deformation paths that preserve the $G$-symmetry. However, two such $G$-SPT states may be able to smoothly deform into each other if we view them as systems with the extended $H$-symmetry and deform them along the paths that preserve the $H$-symmetry. "

To me, it sounds like the two sentences contradict each other.

Q1) If there is an $H$ invariant deformation path to connect the system to a trivial state, then does that not automatically give us a $G$ invariant deformation path?

Q2) Does sentence 2 somehow only apply to the boundary rather than the bulk?