  In this paper we study the second Hochschild cohomology group of the
preprojective algebra of type $D_4$ over an algebraically closed field $K$ of
characteristic 2. We also calculate the second Hochschild cohomology group of a
non-standard algebra which arises as a socle deformation of this preprojective
algebra and so show that the two algebras are not derived equivalent. This
answers a question raised by Holm and Skowro\'nski.
