  We prove a generalization of Tukia's ('85) isomorphism theorem which states
that isomorphisms between geometrically finite groups extend equivariantly to
the boundary. Tukia worked in the setting of real hyperbolic spaces of finite
dimension, and his theorem cannot be generalized as stated to the setting of
CAT($-1$) spaces. We exhibit examples of type-preserving isomorphisms of
geometrically finite subgroups of finite-dimensional rank one symmetric spaces
of noncompact type (ROSSONCTs) whose boundary extensions are not
quasisymmetric. A sufficient condition for a type-preserving isomorphism to
extend to a quasisymmetric equivariant homeomorphism between limit sets is that
one of the groups in question is a lattice, and that the underlying base fields
are the same, or if they are not the same then the base field of the space on
which the lattice acts has the larger dimension. This in turn leads to a
generalization of a rigidity theorem of Xie ('08) to the setting of
finite-dimensional ROSSONCTs.
