The only maximal affine Hadamard families stemming from the Fourier matrix $$F_{12}=\left[\begin{array}{llllllllllll} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & w & w^2 & w^3 & w^4 & w^5 & w^6 & w^7 & w^8 & w^9 & w^{10} & w^{11}\\ 1 & w^2 & w^4 & w^6 & w^8 & w^{10} & 1 & w^2 & w^4 & w^6 & w^8 & w^{10}\\ 1 & w^3 & w^6 & w^9 & 1 & w^3 & w^6 & w^9 & 1 & w^3 & w^6 & w^9 \\ 1 & w^4 & w^8 & 1 & w^4 & w^8 & 1 & w^4 & w^8 & 1 & w^4 & w^8 \\ 1 & w^5 & w^{10} & w^3 & w^8 & w & w^6 & w^{11} & w^4 & w^9 & w^2 & w^7 \\ 1 & w^6 & 1 & w^6 & 1 & w^6 & 1 & w^6 & 1 & w^6 & 1 & w^6 \\ 1 & w^7 & w^2 & w^9 & w^4 & w^{11} & w^6 & w & w^8 & w^3 & w^{10} & w^5 \\ 1 & w^8 & w^4 & 1 & w^8 & w^4 & 1 & w^8 & w^4 & 1 & w^8 & w^4 \\ 1 & w^9 & w^6 & w^3 & 1 & w^9 & w^6 & w^3 & 1 & w^9 & w^6 & w^3 \\ 1 & w^{10} & w^8 & w^6 & w^4 & w^2 & 1 & w^{10} & w^8 & w^6 & w^4 & w^2 \\ 1 & w^{11} & w^{10} & w^9 & w^8 & w^7 & w^6 & w^5 & w^4 & w^3 & w^2 & w \\ \end{array}\right]$$ with $w=\exp(2\pi i/12)$ are $$\begin{aligned} F_{12\Sigma}^{(9)}(a,b,c,d,e,f,g,h,i) &= F_{12}\circ{\rm EXP}\left(i R_{F_{12\Sigma}^{(9)}}(a,b,c,d,e,f,g,h,i)\right),\\ \left(F_{12\Sigma}^{(9)}(a,b,c,d,e,f,g,h,i)\right)^{\rm T} &= F_{12}\circ{\rm EXP}\left(i \left(R_{F_{12\Sigma}^{(9)}}(a,b,c,d,e,f,g,h,i)\right)^{\rm T}\right) \end{aligned}$$ for $\Sigma\in\{A,B,C,D\},$ where $$\begin{aligned} R_{F_{12A}^{(9)}}&(a,b,c,d,e,f,g,h,i)=\\ &\left[\begin{array}{rrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & d & e & \bullet & a & b & c & d & e\\ \bullet & f & \bullet & f & \bullet & f & \bullet & f & \bullet & f & \bullet & f\\ \bullet & g & b & c-a+g & d & e-a+g & \bullet & g & b & c-a+g & d & e-a+g\\ \bullet & h & \bullet & h & \bullet & h & \bullet & h & \bullet & h & \bullet & h\\ \bullet & i & b & c-a+i & d & e-a+i & \bullet & i & b & c-a+i & d & e-a+i\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & d & e & \bullet & a & b & c & d & e\\ \bullet & f & \bullet & f & \bullet & f & \bullet & f & \bullet & f & \bullet & f\\ \bullet & g & b & c-a+g & d & e-a+g & \bullet & g & b & c-a+g & d & e-a+g\\ \bullet & h & \bullet & h & \bullet & h & \bullet & h & \bullet & h & \bullet & h\\ \bullet & i & b & c-a+i & d & e-a+i & \bullet & i & b & c-a+i & d & e-a+i \end{array}\right], \\ \\ \\ R_{F_{12B}^{(9)}}&(a,b,c,d,e,f,g,h,i)=\\ &\left[\begin{array}{rrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & d & e & \bullet & a & b & c & d & e\\ \bullet & f & g & \bullet & f & g & \bullet & f & g & \bullet & f & g\\ \bullet & h & i & c & d-a+h & e-b+i & \bullet & h & i & c & d-a+h & e-b+i\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & d & e & \bullet & a & b & c & d & e\\ \bullet & f & g & \bullet & f & g & \bullet & f & g & \bullet & f & g\\ \bullet & h & i & c & d-a+h & e-b+i & \bullet & h & i & c & d-a+h & e-b+i\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & d & e & \bullet & a & b & c & d & e\\ \bullet & f & g & \bullet & f & g & \bullet & f & g & \bullet & f & g\\ \bullet & h & i & c & d-a+h & e-b+i & \bullet & h & i & c & d-a+h & e-b+i \end{array}\right], \\ \\ \\ R_{F_{12C}^{(9)}}&(a,b,c,d,e,f,g,h,i)=\\ &\left[\begin{array}{rrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & \bullet & d & b & a & \bullet & c & b & d\\ \bullet & e & f & e & \bullet & e & f & e & \bullet & e & f & e\\ \bullet & g & \bullet & c-a+g & \bullet & d-a+g & \bullet & g & \bullet & c-a+g & \bullet & d-a+g\\ \bullet & h & b & h & \bullet & h & b & h & \bullet & h & b & h\\ \bullet & i & f & c-a+i & \bullet & d-a+i & f & i & \bullet & c-a+i & f & d-a+i\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & \bullet & d & b & a & \bullet & c & b & d\\ \bullet & e & f & e & \bullet & e & f & e & \bullet & e & f & e\\ \bullet & g & \bullet & c-a+g & \bullet & d-a+g & \bullet & g & \bullet & c-a+g & \bullet & d-a+g\\ \bullet & h & b & h & \bullet & h & b & h & \bullet & h & b & h\\ \bullet & i & f & c-a+i & \bullet & d-a+i & f & i & \bullet & c-a+i & f & d-a+i \end{array}\right], \\ \\ \\ R_{F_{12D}^{(9)}}&(a,b,c,d,e,f,g,h,i)=\\ &\left[\begin{array}{llllllllllll} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & d & a & \bullet & c & b & a & d & c\\ \bullet & e & \bullet & f & \bullet & e & \bullet & f & \bullet & e & \bullet & f\\ \bullet & g & b & g & d & g & \bullet & g & b & g & d & g\\ \bullet & h & \bullet & c-a+h & \bullet & h & \bullet & c-a+h & \bullet & h & \bullet & c-a+h\\ \bullet & i & b & f-e+i & d & i & \bullet & f-e+i & b & i & d & f-e+i\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & a & b & c & d & a & \bullet & c & b & a & d & c\\ \bullet & e & \bullet & f & \bullet & e & \bullet & f & \bullet & e & \bullet & f\\ \bullet & g & b & g & d & g & \bullet & g & b & g & d & g\\ \bullet & h & \bullet & c-a+h & \bullet & h & \bullet & c-a+h & \bullet & h & \bullet & c-a+h\\ \bullet & i & b & f-e+i & d & i & \bullet & f-e+i & b & i & d & f-e+i \end{array}\right]. \end{aligned}$$ Thus, there are three pairs of cognate families, and one self-cognate family $F_{12A}^{(9)}$.