  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.
