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We provide a criterion for when Hilbert squares of complex projective K3surfaces with Picard number one are strongly ambiguous. This criterion is thesame as [DM, Proposition 3.14], but is obtained by a different method. Inparticular, this enables us to compute the automorphism groups of these Hilbertsquares from a different point of view with respect to [BCNS].

Strongly ambiguous Hilbert squares of projective K3 surfaces with Picard number one