  In this paper we will describe all vector-valued Siegel modular forms of
degree 2 and weight ${\rm Sym}^6({\rm St}) \otimes \det^{k}({\rm St})$ with $k$
odd. These vector-valued forms constitute a module over the ring of classical
Siegel modular forms of degree 2 and even weight and this module turns out to
be free. In order to find generators, we generalize certain Rankin-Cohen
differential operators on triples of classical Siegel modular forms that were
first considered by Ibukiyama and we find a Rankin-Cohen bracket on
vector-valued Siegel modular forms.
