On Morita theory for self-dual modules

Let $G$ be a finite group and let $k$ be a field of characteristic $p$. It is known that a $kG$-module $V$ carries a non-degenerate $G$-invariant bilinear form $b$ if and only if $V$ is self-dual. We show that whenever a Morita bimodule $M$ which induces an equivalence between two blocks $B(kG)$ and $B(kH)$ of group algebras $kG$ and $kH$ is self-dual then the correspondence preserves self-duality. Even more, if the bilinear form on $M$ is symmetric then for $p$ odd the correspondence preserves the geometric type of simple modules. In characteristic 2 this holds also true for projective modules.