On Picent for blocks with normal defect group

We prove that if $b$ is a block of a finite group with normal abelian defect group and inertial quotient a direct product of elementary abelian groups, then $\operatorname{Picent}(b)$ is trivial. We also provide examples of blocks $b$ of finite groups with non-trivial $\operatorname{Picent}(b)$. We even have examples with normal abelian defect group and abelian inertial quotient.