Extension in generalized Orlicz--Sobolev spaces

We study the existence of an extension operator $\Lambda \colon W^{1,\varphi}(\Omega)\to W^{1,\psi}(\mathbb{R}^n)$. We assume that $\varphi \in \Phi_\mathrm{w}(\Omega)$ has generalized Orlicz growth, $\psi \in \Phi_\mathrm{w}(\mathbb{R}^n)$ is an extension of $\varphi$, and that $\Omega\subset\mathbb{R}^n$ is an $(\epsilon,\delta)$-domain. Special cases include the classical constant exponent case, the Orlicz case, the variable exponent case, and the double phase case.