& 0     & 0                 & {\mathbb{Z}}^3             & 0         & {\mathbb{Z}}^2             & {\mathbb{Z}}^5
    \\ \hline
    \mathrm{BDI}    & 0     & {\mathbb{Z}}& 0                 & 0         & {\mathbb{Z}}& 0
    \\ \hline
    \mathrm{D}      & 0     & 0                 & {\mathbb{Z}}& 0         & 0                 & 0
    \\ \hline
    \mathrm{DIII}   & {\mathbb{Z}}& {\mathbb{Z}}^3             & 0                 & {\mathbb{Z}}^5     & 0                 & 0
    \\ \hline
    \mathrm{AII}    & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2           & {\mathbb{Z}}^3             & {\mathbb{Z}}_2^5   & {\mathbb{Z}}^2             & {\mathbb{Z}}^5 
    \\ \hline
    \mathrm{CII}    & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^2           & {\mathbb{Z}}_2^5   & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^5       
    \\ \hline
    \mathrm{C}      & 0     & 0                 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& 0         & {\mathbb{Z}}_2^3           & {\mathbb{Z}}_2^5
    \\ \hline
    \mathrm{CI}     & {\mathbb{Z}}& {\mathbb{Z}}^3             & 0                 & {\mathbb{Z}}^5     & {\mathbb{Z}}_2             & 0 
    \end{array}$$

\[tab:exTd\]

In this paper, we study the structure of the ${\mathbb{Z}}_2$-graded structure algebra extension ${B^0\hookrightarrow B}$ in presence of additional symmetries of the point group, ${g\in G}$, which act on the momenta by $$\label{Og}
U_g^{\phantom{|}}\mathcal{H}(\mathbf{k})U_g^{-1}=\mathcal{H}(O_g\mathbf{k}),$$ where ${O_g\in\mathrm{O}(d)}$ is an orthogonal transformation. This yields an action on the Clifford algebras such that $G$ acts only on the spatial generators $$U_g^{\phantom{|}}\bm{\gamma}U_g^{-1}=O_g^{\mathrm{T}}\bm{\gamma}.$$ However, due to the spin of the electrons, the unitary actions, $U_g$, form a projective representation, $\Pi(G)$, satisfying $$\label{spinor}
U_{g}U_{g'}=\pm U_{g\cdot g'}$$ (see Sec. \[sec:pgs\]). These algebras of spatial and extended 