  Given a suitable action on a complex projective variety X of a non-reductive
affine algebraic group H, this paper considers how to choose a reductive group
G containing H and a projective completion of G x_H X which is a reductive
envelope in the sense of math.AG/0703131. In particular it studies the family
of examples given by moduli spaces of hypersurfaces in the weighted projective
plane P(1,1,2) obtained as quotients by linear actions of the (non-reductive)
automorphism group of P(1,1,2).
