  Consider a finitely generated group $G$ that is relatively hyperbolic with
respect to a family of subgroups $H_1, ..., H_n$. We present an axiomatic
approach to the problem of extending metric properties from the subgroups $H_i$
to the full group $G$. We use this to show that both (weak) finite
decomposition complexity and straight finite decomposition complexity are
extendable properties. We also discuss the equivalence of two notions of
straight finite decomposition complexity.
