$Sp(m)\cdot Sp(1)$ is not the stabilizer of any tensors

Question

Since all groups in Berger's theorem are the stabilizers of a finite number of tensors except possibly the one in the title, I was hoping to receive a confirmation that this is the case. I do not expect it to be the stabilizer of some tensors because a quaternionic Kähler structure cannot be expressed by tensor fields, but any proper motivation is welcome.