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Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and with an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, \emph{always existent} for \emph{any} SPT state in any spacetime dimension $d \geq 2$ of \emph{any} finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary of bulk $d \geq 2$, or more naturally by a topological {emergent} $K$-gauge theory with a global symmetry $G$ on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has \emph{spontaneous symmetry breaking} with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped boundaries/interfaces obtained via \emph{symmetry breaking} (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on \emph{symmetry extension}. More generally, applying our approach to SPT states, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of \emph{symmetry breaking}, \emph{symmetry extension}, and \emph{dynamical gauging}.

Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Lattice (hep-lat); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Algebraic Topology (math.AT); Topological Phases of Matter; TQFT; Quantum Field Theories