  Terminalizations of symplectic quotients are sources of new deformation types
of irreducible symplectic varieties. We classify all terminalizations of
quotients of Hilbert schemes of K3 surfaces or of generalized Kummer varieties,
by finite groups of symplectic automorphisms induced from the underlying K3 or
abelian surface. We determine their second Betti number and the fundamental
group of their regular locus. In the Kummer case, we prove that the
terminalizations have quotient singularities, and determine the singularities
of their universal quasi-\'etale cover. In particular, we obtain at least nine
new deformation types of irreducible symplectic varieties of dimension four.
Finally, we compare our deformation types with those in [FM21; Men22]. The
smooth terminalizations are only three and of K$3^{[n]}$-type, and surprisingly
they all appeared in different places in the literature [Fuj83; Kaw09; Flo22].
