Let us consider 't Hooft anomaly: \begin{eqnarray} Z[A^\lambda]=Z[A]\exp(i\alpha[A,\lambda]), \end{eqnarray} where $A$ is the background $G$-gauge field and $\lambda$ is some $G$-gauge transformation.

We know that a nontrivial factor $\exp(i\alpha[A,\lambda])\neq1$ obstructs $G$-gauging and, if the symmetry $G$ is acted in an on-site manner, such a system cannot be realized in its own dimension(s).

My question is, how can we argue that the anomalous system cannot have a fully gapped spectrum with a unique ground state? In my understanding, a unique gapped ground state must be $G$-singlet, so we expect its partition function should be unambiguous. However, how can a gapless spectrum or a $G$-symmetric (non-symmetry-spontaneously-broken (SSB)) multiple ground states or SSB have an ambiguous partition function?

My second question is, the anomalous symmetry only implies that the symmetry cannot be on-site-ly realized. If the symmetry is non-onsite at the UV scale, we can still have a nontrivial anomaly in the IR field theory. In this case, can we argue that an IR anomaly implies that the system at the UV cannot be gapped by local $G$-symmetric interactions with a unique ground state?