An induced representation of an SO(2N + 1) group has been obtained from a group extension of the SO(2N) Bogoliubov transformation for fermions to a new canonical transformation group. Embedding the SO(2N + 1) group into an SO(2N + 2) group and using the SO(2N + 2)/U(N + 1) coset variables, we develop an extended time dependent Hartree–Bogoliubov (TDHB) theory in which paired and unpaired modes are treated in an equal manner. The extended TDHB theory applicable to both even and odd fermion systems is a time dependent self-consistent field (TDSCF) theory with the same level of the mean field approximation as the usual TDHB theory for even fermion systems. We start from the Hamiltonian of the fermion system which includes, however, the Lagrange multiplier terms to select the physical spinor subspace. The extended TDHB equation is derived from the classical Euler–Lagrange equation of motion for the SO(2N + 2)/U(N + 1) coset variables in the TDSCF. The final form of the extended TDHB equation can be expressed through the variables as the representatives of the paired mode and the unpaired mode. We introduce the quasi anti-commutation relation approximation for the fermion. The parameters included in the Lagrangian multiplier terms are determined under the quasi anti-commutation relation approximation.