Beyond a Question of Markus Linckelmann

In the 2002 Durham Symposium, Markus Linckelmann conjectured the existence of a regular central [Formula: see text]-extension of the full subcategory over the selfcentralizing Brauer pairs of the Frobenius [Formula: see text]-category [Formula: see text] associated with a block [Formula: see text] of defect group [Formula: see text] of a finite group [Formula: see text], which would include, as [Formula: see text]-automorphism groups of the objects, the [Formula: see text]-groups associated with the automizers of the corresponding selfcentralizing Brauer pairs. As a matter of fact, in this question the selfcentralizing Brauer pairs can be replaced by the nilcentralized Brauer pairs, still getting a positive answer. But the condition on the [Formula: see text]-automorphism groups of the objects is not precise enough to guarantee the uniqueness of a solution. This uniqueness depends on the folder structure associated with [Formula: see text], and here we prove the existence and the uniqueness for any folded Frobenius [Formula: see text]-category.