  Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual
$\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest
possible minimum Euclidean weight for a given length. A small number of such
codes is known for lengths greater than or equal to $48.$ The doubling method
is a method for constructing Type II $\mathbb{Z}_4$-codes from a given Type II
$\mathbb{Z}_4$-code. Based on the doubling method, in this paper we develop a
method to construct new extremal Type II $\mathbb{Z}_4$-codes starting from an
extremal Type II $\mathbb{Z}_4$-code of type $4^k$ with an extremal residue
code and length $48, 56$ or $64$. Using this method, we construct three new
extremal Type II $\mathbb{Z}_4$-codes of length $64$ and type $4^{31}2^2$.
Extremal Type II $\mathbb{Z}_4$-codes of length $64$ of this type were not
known before. Moreover, the residue codes of the constructed extremal
$\mathbb{Z}_4$-codes are new best known $[64,31]$ binary codes and the supports
of the minimum weight codewords of the residue code and the torsion code of one
of these codes form self-orthogonal $1$-designs.
