  In the past decade, there has been a systematic investigation of
symmetry-protected topological (SPT) phases in interacting fermion systems.
Specifically, by utilizing the concept of equivalence classes of finite-depth
fermionic symmetric local unitary (FSLU) transformations and the fluctuating
decorated symmetry domain wall picture, a large class of fixed-point wave
functions have been constructed for fermionic SPT (FSPT) phases. Remarkably,
this construction coincides with the Atiyah-Hirzebruch spectral sequence,
enabling a complete classification of FSPT phases. However, unlike bosonic SPT
phases, the stacking group structure in fermion systems proves to be much more
intricate. The construction of fixed-point wave functions does not explicitly
provide this information. In this paper, we employ FSLU transformations to
investigate the stacking group structure of FSPT phases. Specifically, we
demonstrate how to compute stacking FSPT data from the input FSPT data in each
layer, considering both unitary and anti-unitary symmetry, up to 2+1
dimensions. As concrete examples, we explicitly compute the stacking group
structure for crystalline FSPT phases in all 17 wallpaper groups and the
mixture of wallpaper groups with onsite time-reversal symmetry using the
fermionic crystalline equivalence principle. Importantly, our approach can be
readily extended to higher dimensions, offering a versatile method for
exploring the stacking group structure of FSPT phases.
