Extension in generalized Orlicz--Sobolev spaces

We study the existence of an extension operator $\mathrm{\Lambda }\text{\hspace{0.17em}⁣}:W^{1,\phi }(\mathrm{\Omega })\to W^{1,\psi }({\mathbb{R}}^n)$. We assume that $\phi \in {\mathrm{\Phi }}_{\mathrm{w}}(\mathrm{\Omega })$ has generalized Orlicz growth, $\psi \in {\mathrm{\Phi }}_{\mathrm{w}}({\mathbb{R}}^n)$ is an extension of $\phi$, and that $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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.