Quandles with involutions that satisfy certain conditions, called good
involutions, can be used to color non-orientable surface-knots. We use
subgroups of signed permutation matrices to construct non-trivial good
involutions on extensions of odd order dihedral quandles.

For the smallest example of order 6 that is an extension of the three-element
dihedral quandle, various symmetric quandle homology groups are computed, and
applications to the minimal triple point number of surface-knots are given.