  We present a sufficient condition for the punctural Hilbert scheme of length
two of a K3 surface with finite automorphism group to have automorphism group
of infinite order in geometric terms (Theorem 2.1). We then give concrete
examples (Theorem 1.2). We also discuss about Mori dream space (MDS) structures
under an extermal crepant resolution (Theorems 1.2, 5.2, 4.1) from the
viewpoint of automorphisms. These affirmatively answer a question of Doctor
Malte Wandel.
