  A fermionic supersymmetric extension is established for the Gauss-Weingarten
and Gauss-Codazzi equations describing conformally parametrized surfaces
immersed in a Grassmann superspace. An analysis of this extension is performed
using a superspace-superfield formalism together with a supersymmetric version
of a moving frame on a surface. In contrast with the bosonic supersymmetric
extension, the equations of the fermionic supersymmetric Gauss-Codazzi model
resemble the form of the classical equations. Next, a superalgebra of Lie point
symmetries of these equations is determined and a classification of the
one-dimensional subalgebras of this superalgebra into conjugacy classes is
presented. The symmetry reduction method is used to obtain group-invariants,
orbits and reduced systems for three chosen one-dimensional subalgebras. The
explicit solutions of these reduced systems correspond to different surfaces
immersed in a Grassmann superspace. Within this framework for the
supersymmetric version of the Gauss-Codazzi equations a geometrical
interpretation of the results is dicussed.
