Pseudorepresentations not arising from genuine representations

We show that a pseudorepresentation $\mathbf{\text{D}}\text{\hspace{0.17em}⁣}:A[G]\to A$ of a (finite) group $G$ need not arise from a genuine representation, even if one is allowed to extend the ring $A$. This shows that a theorem of the “embedding problem” for residually multiplicity free pseudorepresentations can not be extended to the general setting.