We study 4-dimensional fermionic anomalies with discrete ${\mathbb{Z}}_n$ symmetry, classified by the 5d spin bordism group. We show that only the anomaly from the group-cohomology subclass $\H^5(\mathbb{Z}_n,\U(1))\cong \mathbb{Z}_n$ can be canceled by an anomalous ${\mathbb{Z}}_n$-symmetric 4d ${\mathbb{Z}}_n$-gauge topological quantum field theory (TQFT), while the beyond-group-cohomology anomaly involving generic $A_{\Z_n}p_1$ with Pontryagin class cannot be trivialized by any finite group extension (except $n=2,3$). More generally, we prove that any cocycle $\alpha_d \in \H^d(\mathbb{Z}_n,\U(1))$ in odd spacetime dimension $d\ge 3$ is trivialized by the symmetry extension $ 1 n {n^2} _n , $ and we construct explicitly the corresponding symmetric anomalous boundary TQFT. % As an application, to provide a nonperturbative global anomaly cancellation mechanism for an implication of the structure of the Standard Model (SM), we construct a 4d ${\mathbb{Z}}_N$-gauge TQFT that cancels the mixed discrete baryon-plus-lepton $(𝐁+𝐋)$-gauge-gravitational global anomaly of the generalized SM with $N_f$ families and $N_c$ colors, in the absence of $N_f$ families of "sterile" right-handed neutrinos ${\nu }_R$. In particular, for $d=5$ and $n=3$, a $Spin\times {\mathbb{Z}}_3$-symmetric 4d ${\mathbb{Z}}_3$-gauge TQFT can replace the 3 families of ${\nu }_R$ but still preserve the 3-family SM’s $\Z_{6,{{\bf B} + {\bf L}}}^\rF$ symmetry. In general, a 4d anomalous $\Spin \times_{\Z_2^\rF} \Z_{2 N_f,{{\bf B} + {\bf L}}}^\rF$ symmetric ${\mathbb{Z}}_N$-gauge TQFT can replace the $N_f$ families of ${\nu }_R$, via an appropriate {} construction $$1 \to \mathbb{Z}_N\to \Spin\times \mathbb{Z}_{N N_f}\to \Spin \times_{\Z_2^\rF} \Z_{2 N_f}^\rF \to 1$$ %{$1\to {\mathbb{Z}}_{N_c}\to {\mathbb{Z}}_{N_c{N}_f}\to {\mathbb{Z}}_{N_f}\to 1$} of anomalous topological order . If $N$ and $N_f$ are minimal nonzero positive integers, then we find minimal extensions: $$\{\begin{array}{cccc}N=1,\hfill & N_f\ge 1,\hfill & 2\nmid N_f,\hfill & 3\nmid N_f.\hfill \\ N=3,\hfill & N_f\ge 3,\hfill & 2\nmid N_f,\hfill & 3\mid N_f.\hfill \\ N=4,\hfill & N_f\ge 2,\hfill & 2\mid N_f,\hfill & 3\nmid N_f.\hfill \\ N=12,\hfill & N_f\ge 6,\hfill & 2\mid N_f,\hfill & 3\mid N_f.\hfill \end{array}$$ % The above symmetry extension also constrains that if $N_f$ is odd, then the minimal $N$ is odd. % Further, Witten anomaly constrains the SM such that if $N_f$ is odd, then $N_c$ must be odd. % If and only if $N_c$ and $N_f$ are odd, the anomaly-trivialization symmetry extension construction can coincide with the baryon $\bf B$ to quark $\bf Q$ {} $$ 1 \to \Z_{N_c}\to \Spin \times_{\Z_2^\rF} \Z_{2N_cN_f, {\bf Q} +N_c {\bf L}}\to \Spin \times_{\Z_2^\rF} \Z_{2 N_f,{\bf B +L}}^\rF\to 1. $$ % % If and only if $N_c$ is odd, then SM baryons are fermions. % So only with $N$, $N_c$, and $N_f$ all odd integers, if further assuming $N=N_c$, then $\Z_N$-gauge TQFT can coincide with the $\SU(N_c)$ color gauge group center $Z(\SU(N_c))$. We prove that 3 families and 3 colors, $N=N_f=N_c=3$, is the unique minimal %SM family puzzle solution to have an anomalous ${\mathbb{Z}}_N$-gauge TQFT matching the anomaly of $N_f$ of ${\nu }_R$. % We also prove the identity $A_{{\mathbb{Z}}_3}{p}_1=0mod3$ on oriented manifolds, and $A_{{\mathbb{Z}}_2}{p}_1=0mod2$ on oriented spin manifolds; for $n>3$, the analogous $A_{{\mathbb{Z}}_n}{p}_1=0modn$ is false in general.