Bug in `PreImagesRepresentative` for projective action

[ThomasBreuer]

The following happens in GAP 4.15.1 and in the master branch.

gap> m:= [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] * Z(3)^0;;
gap> G:= Group( m );;
gap> D:= NormedRowVectors( GF(3)^3 );;
gap> hom:= ActionHomomorphism( G, D, OnLines );;
gap> IsInjective( hom );
true
gap> img:= m^hom;
(6,7)(8,11)(9,13)(10,12)
gap> pre:= PreImagesRepresentative( hom, img );
[ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3), 0*Z(3) ], 
  [ 0*Z(3), 0*Z(3), Z(3) ] ]
gap> pre in G;
false

The result matrix pre is correct up to a scalar multiple, but the group G does not contain nontrivial scalar matrices.
The special method for IsProjectiveActionHomomorphism claims that it can compute the preimage but apparently gets it wrong.
The generic method that delegates to an AsGroupGeneralMappingByImages for hom gives the correct result.

[ThomasBreuer]

My interpretation is:
The special PreImagesRepresentative code for IsProjectiveActionHomomorphism uses that the available data about the projective action allow GAP in certain situations to reconstruct the multiplicative coefficients c that occur in OnLines( v, mat ) = c * OnRight( v, mat ), up to one global scalar for mat.
In all other situations, the code calls TryNextMethod.

In order to "determine also the global scalar", the code (LinearActionBasis) tries to use determinants.
But the effect is just that the result is a matrix of determinant one. The example above shows that this does in general not yield a correct result.

I think that the special PreImagesRepresentative method for IsProjectiveActionHomomorphism makes sense only if the Source of the homomorphism contains all scalar matrices in the field of definition, or if the group is contained in SL and if determinants determine the global scalar.r
In this sense LinearActionBasis must do more checks before it provides data that are later on used by PreImagesRepresentative.