  We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs)
as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are
obtained by gauging $G$-equivariant invertible spin-TQFTs, or, in physics
language, gauging the interacting fermionic Symmetry Protected Topological
states (SPTs) with a finite group $G$ symmetry. We use the fact that the latter
are classified by Pontryagin duals to spin-bordism groups of the classifying
space $BG$. We also consider unoriented analogues, that is $G$-equivariant
invertible pin$^\pm$-TQFTs (fermionic time-reversal-SPTs) and their gauging. We
compute these groups for various examples of abelian $G$ using Adams spectral
sequence and describe all corresponding TQFTs via certain bordism invariants in
dimensions 3, 4, and other. This gives explicit formulas for the partition
functions of spin-TQFTs on closed manifolds with possible extended operators
inserted. The results also provide explicit classification of 't Hooft
anomalies of fermionic QFTs with finite abelian group symmetries in one
dimension lower. We construct new anomalous boundary deconfined spin-TQFTs
(surface fermionic topological orders). We explore SPT and SET (symmetry
enriched topologically ordered) states, and crystalline SPTs protected by
space-group (e.g. translation $\mathbb{Z}$) or point-group (e.g. reflection,
inversion or rotation $C_m$) symmetries, via the layer-stacking construction.
