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 Kahler manifold. Explicit anomaly calculations are carried out in the $n=1$ case.
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