Understanding the implications of \(\text{SL}(2,\mathbb Z)\) S-duality for the hyper-multiplet moduli space of type II string theories has led to much progress recently in uncovering D-instanton contributions. In this work, we suggest that the extended duality group \(\text{SL}(3,\mathbb Z)\), which includes both S-duality and Ehlers symmetry, may determine the contributions of D5 and NS5-branes. We support this claim by automorphizing the perturbative corrections to the “extended universal hypermultiplet”, a five-dimensional universal \(\text{SO}(3)\setminus \text{SL}(3,\mathbb R)\) subspace which includes the string coupling, overall volume, Ramond zero-form and six-form and NS axion. Using the non-Abelian Fourier expansion of the Eisenstein series attached to the principal series of \(\text{SL}(3,\mathbb R)\), worked out many years ago by Vinogradov, Takhtajan and Bump, we extract the contributions of \hbox{D(-1)–D5} and NS5-brane instantons, corresponding to the Abelian and non-Abelian coefficients, respectively. In particular, the contributions of \(k\) NS5-branes can be summarized into a vector of wave functions \(\Psi_{k,\ell}\), \(\ell=0,\dots,k-1\), as expected on general grounds. We also point out that for more general models with a symmetric moduli space \(K\setminus G\), the minimal theta series of \(G\) generates an infinite series of exponential corrections of the form required for “small” \hbox{D(-1)–D1-D3-D5–NS5} instanton bound states. As a mathematical spin-off, we make contact with earlier results in the literature about the spherical vectors for the principal series of \(\text{SL}(3,\mathbb R)\) and for minimal representations.