  Let $F$ be an archimedean field, $G$ a divisible ordered abelian group and
$h$ a group exponential on $G$. A triple $(F,G,h)$ is realised in a
non-archimedean exponential field $(K,\exp)$ if the residue field of $K$ under
the natural valuation is $F$ and the induced exponential group of $(K,\exp)$ is
$(G,h)$. We give a full characterisation of all triples $(F,G,h)$ which can be
realised in a model of real exponentiation in the following two cases: i) $G$
is countable. ii) $G$ is of cardinality $\kappa$ and $\kappa$-saturated for an
uncountable regular cardinal $\kappa$ with $\kappa^{<\kappa} = \kappa$.
Moreover, we show that for any o-minimal exponential field $(K, \exp)$
satisfying the differential equation $\exp' = \exp$, its residue exponential
field is a model of real exponentiation.
