I try to extend here some results of the paper "Moduli spaces of decomposable morphisms of sheaves and quotients by non-reductive groups". Let E, F be decomposable sheaves on a smooth projective variety. I find more linearizations of the natural action of the group $\mathop{\rm Aut}\nolimits(E)\times \mathop{\rm Aut}\nolimits(F)$ on $\mathop{\rm Hom}\nolimits(E,F)$ such that a good quotient of the open set of semi-stable morphisms exists. To obtain this, one associates to $\mathop{\rm Hom}\nolimits(E,F)$ another space of morphisms $\mathop{\rm Hom}\nolimits(E',F')$ in such a way that there is a natural bijection between the set of orbits of an open subset of $\mathop{\rm Hom}\nolimits(E,F)$ and the set of orbits of an open subset of $\mathop{\rm Hom}\nolimits(E',F')$. We can in this case deduce the existence of good quotients of some open subsets of $\mathop{\rm Hom}\nolimits(E,F)$ from the existence of good quotients of the corresponding open subsets of $\mathop{\rm Hom}\nolimits(E',F')$.