Realising $\pi_\ast^e R$-algebras by global ring spectra

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory; the global homotopy theory of Schwede. Specifically, for a global ring spectrum $R$, we consider which classes of ring homomorphisms $\eta_\ast\colon\pi_\ast^e R\rightarrow S_\ast$ can be realised by a map $\eta\colon R\rightarrow S$ in the category of global $R$-modules, and what multiplicative structures can be placed on $S$. If $\eta_\ast$ witnesses $S_\ast$ as a projective $\pi_\ast^e R$-module, then such an $\eta$ exists as a map between homotopy commutative global $R$-algebras. If $\eta_\ast$ is in addition \'{e}tale or $S_0$ is a $\mathbb{Q}$-algebra, then $\eta$ can be upgraded to a map of $\mathbb{E}_\infty$-global $R$-algebras or a map of $\mathbb{G}_\infty$-$R$-algebras, respectively. Various global spectra and $\mathbb{E}_\infty$-global ring spectra are then obtained from classical homotopy theoretic and algebraic constructions, with a controllable global homotopy type.