Dear Eamonn,

However, I want to assign the basis to a variable
back within GAP and work with its vectors.

is the function 'GapObject' for MeatAxe objects what you need? For
example:

gap> mat1 := MeatAxeMat( PermutationMat( (1,2,3), 3, GF(2) ), GF(2) );;
gap> mat2 := MeatAxeMat( PermutationMat( (1,2), 3, GF(2) ), GF(2) );;
gap> m := NaturalModule( Algebra( GF(2), [ mat1, mat2 ] ) );;
gap> l := Lattice(m);;
gap> b := GeneratorsSubmodule( l, 3 );
MeatAxeMat( "/var/tmp/tmp.014092/o/g.s2", GF(2), [ 2, 3 ] )
gap> Display(b);
MeatAxe.Matrix := [
[1,0,1],
[0,1,1]
]*Z(2);
gap> GapObject(b);
[ [ Z(2)^0, 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, Z(2)^0 ] ]

best wishes
Frank