  In both ${\cal N}=1$ and ${\cal N}=2$ supersymmetry, it is known that
$\mathsf{Sp}(2n, {\mathbb R})$ is the maximal duality group of $n$ vector
multiplets coupled to chiral scalar multiplets $\tau (x,\theta) $ that
parametrise the Hermitian symmetric space $\mathsf{Sp}(2n, {\mathbb R})/
\mathsf{U}(n)$. If the coupling to $\tau$ is introduced for $n$ superconformal
gauge multiplets in a supergravity background, the action is also invariant
under super-Weyl transformations. Computing the path integral over the gauge
prepotentials in curved superspace leads to an effective action $\Gamma [\tau,
\bar \tau]$ with the following properties: (i) its logarithmically divergent
part is invariant under super-Weyl and rigid $\mathsf{Sp}(2n, {\mathbb R})$
transformations; (ii) the super-Weyl transformations are anomalous upon
renormalisation. In this paper we describe the ${\cal N}=1$ and ${\cal N}=2$
locally supersymmetric "induced actions" which determine the logarithmically
divergent parts of the corresponding effective actions. In the ${\cal N}=1$
case, superfield heat kernel techniques are used to compute the induced action
of a single vector multiplet $(n=1)$ coupled to a chiral dilaton-axion
multiplet. We also describe the general structure of ${\cal N}=1$ super-Weyl
anomalies that contain weight-zero chiral scalar multiplets $\Phi^I$ taking
values in a K\"ahler manifold. Explicit anomaly calculations are carried out in
the $n=1$ case.
